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This paper deals with approximate solutions of an optimization problem with interval-valued objective function. Four types of approximate solution concepts of the problem are proposed by considering the partial ordering LU on the set of all closed and bounded intervals. We show that these solutions exist under very weak conditions. Under suitable constraint qualifications, we derive Karush-Kuhn-Tucker necessary and sufficient optimality conditions for convex interval-valued optimization problems. [ABSTRACT FROM AUTHOR]

Evolutionary multi-objective optimization (EMO) algorithms are actively used for answering optimization problems with multiple contradictory objectives and scheming interpretable and precise real-time applications. A majority of existing EMO algorithms performs better on two or three objectives non-dominated problems; however, they meet complications in managing and maintaining a set of optimal solutions to multi-objective optimization problems. This paper proposes a search space-based multi-objective evolutionary algorithm (SSMOEA) for multi-objective optimization problems. To accomplish the potential of the search space-based method for solving multi-objective optimization problems and to reinforce the selection procedure toward the ideal direction while sustaining an extensive and uniform distribution of solutions is our key objective. To the best of our knowledge, this paper is the first attempt to propose a search space-based multi-objective evolutionary algorithm for multi-objective optimization. The experimental setup used showed that the proposed algorithm is good and competitive in comparison to the existing EMO algorithms from the viewpoint of finding a scattered and estimated solution set in multi-objective optimization problems. SSMOEA can achieve a good trade-off between search space convergence and search space diversity in the appropriate experimental setup. [ABSTRACT FROM AUTHOR]

Detail enhancement algorithms are important for raw infrared images to improve their overall contrast and highlight important information in them. To solve the problems that current algorithms like GF&DDE have, an improved adaptive detail enhancement algorithm for infrared images based on a guided image filter is proposed in this paper. It chooses the threshold for the base layer image adaptively according to the histogram statistical information and adjusts the mapping range of the histograms according to the dynamic range of the image. Besides, the detail layer is handled by a simpler adaptive gain control method to achieve the good detail enhancement effect. Finally, the base layer and the detail are merged according to the approximate proportion of the background and the details. Experimental results show that the proposed algorithm can adaptively and efficiently enhance different dynamic range images in different scenarios. Moreover, this algorithm has high real-time performance. [ABSTRACT FROM AUTHOR]

A typical interpolation strategy for line segments consists of a transition scheme, a Look-ahead ACC/DEC scheduling, and an interpolation algorithm. In these three parts, the main computation occurs in the first and second part. Some research work has been carried out to decrease the computation in the previous literatures, but these methods occupy a lot of computing resources for the optimization process during the calculation of transition curve parameters and feed rates. Consequently, the computational efficiency of interpolation strategy is greatly reduced. To deal with the issue, a real-time interpolation strategy is proposed in this paper. In the transition scheme, a Bézier curve is utilized to smooth the line segments. Based on the relationship among the approximation error, the approximation radius, and the transition curve, the curve can be directly generated when the approximation error is given. In the ACC/DEC scheduling, a 3-segment feed rate profile with jerk continuity is constructed. Meanwhile, a Look-ahead planning based on Backward Scanning and Forward Revision (BSFR) algorithm is utilized to eliminate redundant computation. Compared with Zhao’s and Shi’s strategy, the proposed strategy has the merits of C2 and G2 continuity for the tool path, jerk continuity for the tool movement, and distinguished real-time performance for interpolation. The experiments of 3D pentagram and 2D butterfly are carried out with different strategies and their results demonstrate that the interpolation efficiency can be greatly improved with the proposed strategy. [ABSTRACT FROM AUTHOR]

In this paper, a three-term conjugate gradient algorithm is developed for solving large-scale unconstrained optimization problems. The search direction at each iteration of the algorithm is determined by rectifying the steepest descent direction with the difference between the current iterative points and that between the gradients. It is proved that such a direction satisfies the approximate secant condition as well as the conjugacy condition. The strategies of acceleration and restart are incorporated into designing the algorithm to improve its numerical performance. Global convergence of the proposed algorithm is established under two mild assumptions. By implementing the algorithm to solve 75 benchmark test problems available in the literature, the obtained results indicate that the algorithm developed in this paper outperforms the existent similar state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

The computational budget for designing and optimizing engineering systems has been tightened by the growth in the number of technology alternatives to satisfy the increasingly demanding goals in modern systems and the use of sophisticated, but also computationally burdensome, simulation tools to quantitatively choose the best technology alternative more accurately. This paper develops a new Gaussian process metamodel, which leverages trends in engineering objectives that are similar across derivative concepts, as opposed to current approximation methods that handle concepts independently. This metamodel is based on a muitifidelity approach that enhances a new concept prediction with observations of a previous concept with similar trends. The influence of the sizes of the new and the old concept training sets in the performance of the proposed metamodel is explored in this paper. The proposed multifidelity metamodel is tested against traditional independent surrogates in approximating the engine shaft horsepower of a UH-60A modification with a Fenestron tail, where observations from the UH-60A with a conventional tail are available. The proposed metamodels are up to two times as accurate as independent concept modeling for small new concept training sets; then, a multiobjective efficient global optimization algorithm is applied to the proposed metamodel to search for minimal UH-60A power consumption in hover and cruise. This results in Pareto-optimal solutions that are 10% more efficient than the UH-60A baseline. [ABSTRACT FROM AUTHOR]

We study the max-min fair allocation problem in which a set of m indivisible items are to be distributed among n agents such that the minimum utility among all agents is maximized. In the restricted setting, the utility of each item j on agent i is either 0 or some non-negative weight wj. For this setting, Asadpour et al. (ACM Trans Algorithms 8(3):24, 2012) showed that a certain configuration-LP can be used to estimate the optimal value to within a factor of 4+δ, for any δ>0, which was recently extended by Annamalai et al. (in: Indyk (ed) Proceedings of the twenty-sixth annual ACMSIAM symposium on discrete algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015) to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani (SIGecom Exch 5(3):11-18, 2005) showed that it is NP-hard to approximate the problem within any ratio smaller than 2. In this paper we consider the (1,ϵ)-restricted max-min fair allocation problem in which each item j is either heavy (wj=1) or light (wj=ϵ), for some parameter ϵ∈(0,1). We show that the (1,ϵ)-restricted case is also NP-hard to approximate within any ratio smaller than 2. Using the configuration-LP, we are able to estimate the optimal value of the problem to within a factor of 3+δ, for any δ>0. Extending this idea, we also obtain a quasi-polynomial time (3+4ϵ)-approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as ϵ tends to 0, the approximation ratio of our polynomial-time algorithm approaches 3+22≈5.83. [ABSTRACT FROM AUTHOR]

In this paper, we significantly extend the applicability of state-of-the-art ELDP (equations for linearizing discrete product terms) method by providing a new linearization to handle more complicated non-linear terms involving both of discrete and bounded continuous variables. A general class of “representable programming problems” is formally proposed for a much wider range of engineering applications. Moreover, by exploiting the logarithmic feature embedded in the discrete structure, we present an enhanced linear reformulation model which requires half an order fewer equations than the original ELDP. Computational experiments on various engineering design problems support the superior computational efficiency of the proposed linearization reformulation in solving engineering optimization problems with discrete and bounded continuous variables. [ABSTRACT FROM AUTHOR]

In this contribution, a new framework for -optimal reduction is presented, motivated by the local nature of both (tangential) interpolation and -optimal approximations. The main advantage is given by a decoupling of the cost of reduction from the cost of optimization, resulting in a significant speedup in -optimal reduction. In addition, a middle-sized surrogate model is produced at no additional cost and can be used e.g. for error estimation. Numerical examples illustrate the new framework, showing its effectiveness in producing -optimal reduced models at a far lower cost than conventional algorithms. Detailed discussions and optimality proofs are presented for applying this framework to the reduction of multiple-input, multiple-output linear dynamical systems. The paper ends with a brief discussion on how this framework could be extended to other system classes, thus indicating how this truly is a general framework for interpolatory reduction. [ABSTRACT FROM AUTHOR]

The number of formal concepts generated from the input context is an important parameter in the cost functions of concept formation algorithms. The calculation of concept count for any arbitrary context is a hard, NP-complete problem and only rough approximation methods can be found in the literature to solve this problem. This paper introduces an efficient numerical approximation algorithm for contexts where attribute probabilities are independent from the objects instances. The preconditions required by the approximation method are usually met in the FCA applications, thus the proposed method provides an efficient tool for practical complexity analysis, too. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem governed by a regularized fracture model using a phase-field technique. To avoid the nondifferentiability due to the irreversibility constraint on the fracture growth, the phase-field fracture model is relaxed using a penalization approach. The existence of a solution to the penalized fracture model is shown, and the existence of at least one solution for the regularized optimal control problem is established. Moreover, the linearized fracture model is considered and used to establish first order necessary conditions as well as to discuss QP-approximations to the nonlinear optimization problem. A numerical example suggests that these can be used to obtain a fast convergent algorithm. [ABSTRACT FROM AUTHOR]

An edge-bicolored graph G = ( V , R ∪ B ) is called Red/Blue-split graph if there exists a partition I B and I R of V such that I B and I R are independent sets in ( V , B ) and ( V , R ) , respectively. Red/Blue-split graphs generalize several well studied graph classes including split graphs, bipartite graphs and König–Egerváry graphs. In this paper we consider the algorithmic complexity of various optimization problems like minimum edge (or vertex) deletion and maximum edge (or vertex) induced subgraph related to Red/Blue split graphs. All these problems are NP -hard and thus we look at them from algorithmic paradigms that are meant for coping with NP -hardness. We obtain various hardness as well as algorithmic results for these problems in the realm of approximation algorithms and parameterized complexity. The main tool we use to obtain all our results is polynomial time transformations between appropriate problems. On the way, we also resolve some problems related to inapproximability about certain optimization problems mentioned by Korach et al. (2006) [17] . [ABSTRACT FROM AUTHOR]

In this paper, an efficient method based on the sequential quadratic programming (SQP) algorithm for the pattern synthesis of concentric rings antenna arrays to reduce side lobe levels (SLL) by controlling only the amplitude excitations of each array element was presented. SQP algorithm is a deterministic method of multi-objective optimization. It is the most widely used for the resolution of nonlinear problems. It consists of transforming the nonlinear problem to sequence of quadratic sub-problems by using a quadratic approximation of the Lagrangian function. In order to illustrate the performance of the proposed method, several examples of planar concentric rings antenna array patterns with almost one-half wavelength spaced isotropic elements to reduce the SLL and to norrow the half power beam width are presented. The results shown in this paper and compared with those obtained by the application of uniform, Hamming and Blackman weighting, demonstrate the effectiveness of this optimization method based on the SQP algorithm. [ABSTRACT FROM AUTHOR]

The research on the triangle packing problem has important theoretic significance, which has broad application prospects in material processing, network resource optimization, and so forth. Generally speaking, the orientation of the triangle should be limited in advance, since the triangle packing problem is NP-hard and has continuous properties. For example, the polygon is not allowed to rotate; then, the approximate solution can be obtained by optimization method. This paper studies the triangle packing problem by a new kind of method. Such concepts as angle region, corner-occupying action, corner-occupying strategy, and edge-conjoining strategy are presented in this paper. In addition, an edge-conjoining and corner-occupying algorithm is designed, which is to obtain an approximate solution. It is demonstrated that the proposed algorithm is highly efficient, and by the time complexity analysis and the analogue experiment result is found. [ABSTRACT FROM AUTHOR]

This paper presents a methodology for constrained efficient global optimization (EGO) using support vector machines (SVMs). While the objective function is approximated using Kriging, as in the original EGO formulation, the boundary of the feasible domain is approximated explicitly as a function of the design variables using an SVM. Because SVM is a classification approach and does not involve response approximations, this approach alleviates issues due to discontinuous or binary responses. More importantly, several constraints, even correlated, can be represented using one unique SVM, thus considerably simplifying constrained problems. In order to account for constraints, this paper introduces an SVM-based 'probability of feasibility' using a new Probabilistic SVM model. The proposed optimization scheme is constituted of two levels. In a first stage, a global search for the optimal solution is performed based on the 'expected improvement' of the objective function and the probability of feasibility. In a second stage, the SVM boundary is locally refined using an adaptive sampling scheme. An unconstrained and a constrained formulation of the optimization problem are presented and compared. Several analytical examples are used to test the formulations. In particular, a problem with 99 constraints and an aeroelasticity problem with binary output are presented. Overall, the results indicate that the constrained formulation is more robust and efficient. [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses the toll design problem of finding the toll locations and levels in a congestion pricing scheme, which minimize the total travel time and the toll-point cost (set-up and operational costs of the toll collecting facilities). Road users in the network are assumed to be distributed according to the principle of user equilibrium, with the demand assumed to be fixed and given a priori. The toll design problem is commonly formulated as a non-linear program, which in general is non-convex and non-smooth, and thus difficult to solve for a global optimum. In this paper, the toll design problem is approximated by a mixed integer linear program (MILP), which can be solved to its globally optimal solution. The MILP also gives a lower bound estimation of the original non-linear problem, and the accuracy of the approximation is improved by iteratively updating the MILP. To demonstrate the approach, we apply the algorithm to two networks: a smaller network with 18 links and 4 OD-pairs to illustrate the properties of the approach, and the Sioux Falls network with 87 links and 30 OD-pairs to demonstrate the applicability of the approach. [Copyright &y& Elsevier]

Abstract: Extreme learning machine (ELM) is one of the most popular and important learning algorithms. It comes from single-hidden-layer feedforward neural networks. It has been proved that ELM can achieve better performance than support vector machine (SVM) in regression and classification. In this paper, mathematically, with regression problem, the step 3 of ELM is studied. First of all, the equation are reformulated as an optimal model. With the optimality, the necessary conditions of optimal solution are presented. The equation is replaced by . We can prove that the latter must have one solution at least. Second, optimal approximation solution is discussed in cases of is column full rank, row full rank, neither column nor row full rank. In the last case, the rank-1 and rank-2 methods are used to get optimal approximation solution. In theory, this paper present a better algorithm for ELM. [Copyright &y& Elsevier]

Orthogonal Matching Pursuit (OMP) is an effective solution to sparse approximation based on redundant dictionary, but there are plenty of matrix-vector multiplications in order to seek for the most matching atom. This full search scheme can consume much CPU time, especially for high dimension data such as image. Although some existing schemes proposed some accelerating methods by exploiting the property of atoms in parametric dictionary, they cannot be extended to other dictionaries, especially learned dictionary. Considering statistical nature of atoms in learned dictionary, this paper proposes a novel method which utilizes modified K-mean algorithm to cluster all the atoms in the learned dictionary while sufficiently considering the inner product operation between every two atoms, determines the class in which the desired atom locates and finally find it in the class. Some analyses and experiments have shown the success of the method. In addition, this paper gives some empirical analysis of the effect of related parameters of indexed dictionary. [ABSTRACT FROM AUTHOR]

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we focus on the edit distance between two given strings where block-edit operations are allowed and better fitting to the human natural edit behaviors. Previous results showed that this problem is NP-hard when block moves are allowed. Various approximations to this problem have been proposed in recent years. However, this problem can be solved by the polynomial-time optimization algorithms if some reasonable restrictions are applied. The restricted variations which we consider involve character insertions, character deletions, block copies and block deletions. In this paper, three problems are defined with different measuring functions, which are , and . Then we show that with some preprocessing, the minimum block edit distances of these three problems can be obtained by dynamic programming in , and time, respectively, where n and m are the lengths of the two input strings. [Copyright &y& Elsevier]

Abstract: In this paper we use the penalty approach in order to study minimization problems with mixed constraints in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property holds and is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints. [Copyright &y& Elsevier]

Both average-case complexity and the study of the approximability of NP optimisation problems are well established and active fields of research. Many results concerning the average behaviour of approximation algorithms for NP optimisation problems exist, both with respect to their running time and their performance ratio, but a theoretical framework to examine their structural properties with respect to their average-case approximability is not yet established. With this paper we hope to fill the gap and provide not only the necessary definitions, but show that: 1. The class of NP optimisation problems with p-computable input distributions has complete problems with respect to an average-approximability preserving reduction. 2. The average-time variants of worst-case approximation classes form a strict hierarchy if NP is not easy on average. By linking average-ratio approximation algorithms to polynomial-time algorithms schemes, we can prove similar hierarchy results for the average-ratio versions of the approximation classes. This is done under the premise that not all NP-problems with p-computable input distributions have polynomial-time algorithm schemes. 3. The question whether NP is easy on average is equivalent to the question whether every NP optimisation problem with a p-computable input distribution has an average polynomial-time approximation scheme. An extended abstract of this paper appeared in [28]. [ABSTRACT FROM AUTHOR]

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of these problems which can be transformed to index one control problems. For this class of higher index DAEs, we derive first-order approximations and adjoint equations for the functionals defining the problem. These adjoint equations are then used to state, in the accompanying paper, the necessary optimality conditions in the form of a weak maximum principle. The constructive way used to prove these optimality conditions leads to globally convergent algorithms for control problems with state constraints and defined by higher index DAEs. [ABSTRACT FROM AUTHOR]

The paper discusses several versions of the method of shortest residuals, a specific variant of the conjugate gradient algorithm, first introduced by Lemaréchal and Wolfe and discussed by Hestenes in a quadratic case. In the paper we analyze the global convergence of the versions considered. Numerical comparison of these versions of the method of shortest residuals and an implementation of a standard Polak–Ribière conjugate gradient algorithm is also provided. It supports the claim that the method of shortest residuals is a viable technique, competitive to other conjugate gradient algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a novel learning algorithm for efficient construction of the radial basis function (RBF) networks that can deliver the same level of accuracy as the support vector machines (SVMs) in data classification applications. The proposed learning algorithm works by constructing one RBF subnetwork to approximate the probability density function of each class of objects in the training data set. With respect to algorithm design, the main distinction of the proposed learning algorithm is the novel kernel density estimation algorithm that features an average time complexity of O(n log n), where vi is the number of samples in the training data set. One important advantage of the proposed learning algorithm, in comparison with the SVM, is that the proposed learning algorithm generally takes far less time to construct a data classifier with an optimized parameter setting. This feature is of significance for many contemporary applications, in particular, for those applications in which new objects are continuously added into an already large database. Another desirable feature of the proposed learning algorithm is that the RBF networks constructed are capable of carrying out data classification with more than two classes of objects in one single run. In other words, unlike with the SVM, there is no need to resort to mechanisms such as one-against-one or one-against-all for handling datasets with more than two classes of objects. The comparison with SVM is of particular interest, because it has been shown in a number of recent studies that SVM generally are able to deliver higher classification accuracy than the other existing data classification algorithms. As the proposed learning algorithm is instance-based, the data reduction issue is also addressed in this paper. One interesting observation in this regard is that, for all three data sets used in data reduction experiments, the number of training samples remaining after a naïve data reduction mechanism is applied is quite close to the number of support vectors identified by the SVM software. This paper also compares the performance of the RBF networks constructed with the proposed learning algorithm and those constructed with a conventional cluster-based learning algorithm. The most interesting observation learned is that, with respect to data classification, the distributions of training samples near the bound- aries between different classes of objects carry more crucial information than the distributions of samples in the inner parts of the clusters. [ABSTRACT FROM AUTHOR]

Various papers have analyzed the noisy optimization of convex functions. This analysis has been made according to several criteria used to evaluate the performance of algorithms: uniform rate, simple regret and cumulative regret. We propose an iterative optimization framework, a particular instance of which, using Hessian approximations, provably (i) reaches the same rate as Kiefer–Wolfowitz algorithm when the noise has constant variance, (ii) reaches the same rate as Evolution Strategies when the noise variance decreases quadratically as a function of the simple regret, (iii) reaches the same rate as Bernstein-races optimization algorithms when the noise variance decreases linearly as a function of the simple regret. [ABSTRACT FROM AUTHOR]

Being the unbalanced version of the famous Min s – t Cut problem, the Min k -Size s – t Cut problem asks to find a k -size s – t cut with the minimum capacity, where a k -size s – t cut means an s – t cut with its s -side having size at most k . This problem is fundamental and has extensive applications, especially in community identification in social and information networks. It is known that the Min k -Size s – t Cut problem is NP-hard and can be approximated within O ( log ⁡ n ) , where n is the number of vertices in the input graph. In this paper, we give a new approximation algorithm for the Min k -Size s – t Cut problem based on the parametric flow technique. The algorithm is very simple and has only three lines to state. Its approximation ratio is k + 1 k + 1 − k ⁎ , where k ⁎ is the size of the s -side of an optimal solution. [ABSTRACT FROM AUTHOR]

For two positive integers m , k and a connected graph G = ( V , E ) with a nonnegative vertex weight function w , the balanced m -connected k -partition problem, denoted as BC m P k , is to find a partition of V into k disjoint nonempty vertex subsets ( V 1 , V 2 , … , V k ) such that each G [ V i ] (the subgraph of G induced by V i ) is m -connected, and min 1 ≤ i ≤ k ⁡ { w ( V i ) } is maximized. The optimal value of BC m P k on graph G is denoted as β m ⁎ ( G , k ) , that is, β m ⁎ ( G , k ) = max ⁡ min 1 ≤ i ≤ k ⁡ { w ( V i ) } , where the maximum is taken over all m -connected k -partition of G . In this paper, we study the BC 2 P k problem on interval graphs, and obtain the following results. (1) For k = 2 , a 4/3-approximation algorithm is given for BC 2 P 2 on 4-connected interval graphs. (2) In the case that there exists a vertex v with weight at least W / k , where W is the total weight of the graph, we prove that the BC 2 P k problem on a 2 k -connected interval graph G can be reduced to the BC 2 P k − 1 problem on the ( 2 k − 1 ) -connected interval graph G − v . In the case that every vertex has weight at most W / k , we prove a lower bound β 2 ⁎ ( G , k ) ≥ W / ( 2 k − 1 ) for 2 k -connected interval graph G . (3) Assuming that weight w is integral, a pseudo-polynomial time algorithm is obtained. Combining this pseudo-polynomial time algorithm with the above lower bound, a fully polynomial time approximation scheme (FPTAS) is obtained for the BC 2 P k problem on 2 k -connected interval graphs. [ABSTRACT FROM AUTHOR]

It is generally acknowledged that the conjugate gradient (CG) method achieves global convergence—with at most a linear convergence rate—because CG formulas are generated by linear approximations of the objective functions. The quadratically convergent results are very limited. We introduce a new PRP method in which the restart strategy is also used. Moreover, the method we developed includes not only n-step quadratic convergence but also both the function value information and gradient value information. In this paper, we will show that the new PRP method (with either the Armijo line search or the Wolfe line search) is both linearly and quadratically convergent. The numerical experiments demonstrate that the new PRP algorithm is competitive with the normal CG method. [ABSTRACT FROM AUTHOR]

Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. This paper describes the fundamentals of the coordinate descent approach, together with variants and extensions and their convergence properties, mostly with reference to convex objectives. We pay particular attention to a certain problem structure that arises frequently in machine learning applications, showing that efficient implementations of accelerated coordinate descent algorithms are possible for problems of this type. We also present some parallel variants and discuss their convergence properties under several models of parallel execution. [ABSTRACT FROM AUTHOR]

In a two-stage robust covering problem, one of several possible scenarios will appear tomorrow and require to be covered, but costs are higher tomorrow than today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? We consider the $$k$$ -robust model where the possible scenarios tomorrow are given by all demand-subsets of size $$k$$ . In this paper, we give the following simple and intuitive template for $$k$$ -robust covering problems: having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat. We show that this template gives good approximation algorithms for $$k$$ -robust versions of many standard covering problems: set cover, Steiner tree, Steiner forest, minimum-cut and multicut. Our $$k$$ -robust approximation ratios nearly match the best bounds known for their deterministic counterparts. The main technical contribution lies in proving certain net-type properties for these covering problems, which are based on dual-rounding and primal-dual ideas; these properties might be of some independent interest. As a by-product of our techniques, we also get algorithms for max-min problems of the form: ' given a covering problem instance, which $$k$$ of the elements are costliest to cover?' For the problems mentioned above, we show that their $$k$$ -max-min versions have performance guarantees similar to those for the $$k$$ -robust problems. [ABSTRACT FROM AUTHOR]

Abstract: This article considers recovery of signals that are sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame from undersampled data corrupted with additive noise. We show that the properly constrained -analysis optimization problem, called analysis Dantzig selector, stably recovers a signal which is nearly sparse in terms of a tight frame provided that the measurement matrix satisfies a restricted isometry property adapted to the tight frame. As a special case, we consider the Gaussian noise. Further, under a sparsity scenario, with high probability, the recovery error from noisy data is within a log-like factor of the minimax risk over the class of vectors which are at most s sparse in terms of the tight frame. Similar results for the analysis LASSO are shown. The above two algorithms provide guarantees only for noise that is bounded or bounded with high probability (for example, Gaussian noise). However, when the underlying measurements are corrupted by sparse noise, these algorithms perform suboptimally. We propose new algorithms for reconstructing signals that are nearly sparse in terms of a tight frame in the presence of bounded noise combined with sparse noise, and present corresponding recovery guarantees. The analysis in this paper is based on the restricted isometry property adapted to a tight frame, which is a natural extension to the standard restricted isometry property. [Copyright &y& Elsevier]

Abstract: This paper presents a preconditioner for non-overlapping Schwarz methods applied to the Helmholtz problem. Starting from a simple analytic example, we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain. The preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains, and can be interpreted as a coarse grid for the domain decomposition method. The resulting algorithm is shown to converge very fast, independently of the number of subdomains and frequency. The preconditioner has the advantage that, like the original Schwarz algorithm, it can be implemented as a matrix-free routine, with no additional preprocessing. [Copyright &y& Elsevier]

The limited memory BFGS method (L-BFGS) is an adaptation of the BFGS method for large-scale unconstrained optimization. However, The L-BFGS method need not converge for nonconvex objective functions and it is inefficient on highly ill-conditioned problems. In this paper, we proposed a regularization strategy on the L-BFGS method, where the used regularization parameter may play a compensation role in some sense when the condition number of Hessian approximation tends to become ill-conditioned. Then we proposed a regularized L-BFGS method and established its global convergence even when the objective function is nonconvex. Numerical results show that the proposed method is efficient. [ABSTRACT FROM AUTHOR]

We construct a neural network based on smoothing approximation techniques and projected gradient method to solve a kind of sparse reconstruction problems. Neural network can be implemented by circuits and can be seen as an importantmethod for solving optimization problems, especially large scale problems. Smoothing approximation is an efficient technique for solving nonsmooth optimization problems. We combine these two techniques to overcome the difficulties of the choices of the step size in discrete algorithms and the item in the set-valued map of differential inclusion. In theory, the proposed network can converge to the optimal solution set of the given problem. Furthermore, some numerical experiments show the effectiveness of the proposed network in this paper. [ABSTRACT FROM AUTHOR]

Abstract: The problem of identifying a fixed-order FIR approximation of linear systems with unknown structure, assuming that both input and output measurements are subjected to quantization, is dealt with in this paper. A fixed-order FIR model providing the best approximation of the input–output relationship is sought by minimizing the worst-case distance between the output of the true system and the modeled output, for all possible values of the input and output data consistent with their quantized measurements. The considered problem is firstly formulated in terms of robust optimization. Then, two different algorithms to compute the optimum of the formulated problem by means of linear programming techniques are presented. The effectiveness of the proposed approach is illustrated by means of a simulation example. [Copyright &y& Elsevier]

In this paper, as an extension of a previous study, an improved approximation for the Gaussian Q-function is presented. The nonlinear least squares algorithm is employed to optimize the coefficients of the proposed approximation. The accuracy of the presented approximation is evaluated using extensive computer simulations. Results show that the proposed approximation has superior accuracy in high arguments' region when compared to the performance of other approaches introduced in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new nonmonotone inexact line search rule is proposed and applied to the trust region method for unconstrained optimization problems. In our line search rule, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of the current objective function value . We can obtain a larger stepsize in each line search procedure and possess nonmonotonicity when incorporating the nonmonotone term into the trust region method. Unlike the traditional trust region method, the algorithm avoids resolving the subproblem if a trial step is not accepted. Under suitable conditions, global convergence is established. Numerical results show that the new method is effective for solving unconstrained optimization problems. [ABSTRACT FROM AUTHOR]

Based on dynamic programming approach we design algorithms for sequential optimization of exact and approximate decision rules relative to the length and coverage [3, 4]. In this paper, we use optimal rules to construct classifiers, and study two questions: (i) which rules are better from the point of view of classification - exact or approximate; and (ii) which order of optimization gives better results of classifier work: length, length+coverage, coverage, or coverage+length. Experimental results show that, on average, classifiers based on exact rules are better than classifiers based on approximate rules, and sequential optimization (length+coverage or coverage+length) is better than the ordinary optimization (length or coverage). [ABSTRACT FROM AUTHOR]

In this paper, the least-squares solutions of constrained inverse eigenvalue problem and the associated optimal approximation problem are considered. We get the general expression of the least-squares solutions of the constrained inverse eigenvalue problem, obtain the representation of the unique optimal approximation solution in the least-squares solutions set, and give the algorithm and corresponding computational examples. [ABSTRACT FROM PUBLISHER]

This paper presents an active-set algorithm for large-scale optimization that occupies the middle ground between sequential quadratic programming and sequential linear-quadratic programming methods. It consists of two phases. The algorithm first minimizes a piecewise linear approximation of the Lagrangian, subject to a linearization of the constraints, to determine a working set. Then, an equality constrained subproblem based on this working set and using second derivative information is solved in order to promote fast convergence. A study of the local and global convergence properties of the algorithm highlights the importance of the placement of the interpolation points that determine the piecewise linear model of the Lagrangian. [ABSTRACT FROM AUTHOR]

Abstract: The solution of time-dependent PDE-constrained optimization problems subject to unsteady flow equations presents a challenge to both algorithms and computing power. In this paper we present an all-at-once approach where we solve for all time-steps of the discretized unsteady Stokes problem at once. The most desirable feature of this approach is that for all steps of an iterative scheme we only need approximate solutions of the discretized Stokes operator. This leads to an efficient scheme which exhibits mesh-independent behaviour. [Copyright &y& Elsevier]

Abstract: This paper presents new graph-cut based optimization algorithms for image processing problems. Popular graph-cut based algorithms give approximate solutions and are based on the concept of partition move. The main contribution of this work consists in proposing novel partition moves called multilabel moves to minimize Markov random field (MRF) energies with convex prior and any likelihood energy functions. These moves improve the optimum quality of the state-of-the-art approximate minimization algorithms while controlling the memory need of the algorithm at the same time. Thus, the two challenging problems, improving local optimum quality and reducing required memory for graph construction are handled with our approach. These new performances are illustrated on some image processing experiments, such as image restoration and InSAR phase unwrapping. [Copyright &y& Elsevier]

Abstract: Systematic errors inevitably occur during the acquisition and processing of geophysical data because of measurement and discretization noise, as well as incorrect geophysical forward modeling, among other problems. Such errors commonly cause systematic degradations, including over-smoothing and decreased resolution of estimated geophysical models. In this paper, the relationship between systematic errors and an estimated geophysical estimated model is analyzed. A convolution model for systematic degradations is also derived. On the basis of the convolution model, we suggest using the inverse method to reduce systematic degradations and enhance estimated geophysical models. Accordingly, we propose a geophysical model enhancement algorithm based on blind deconvolution. The algorithm uses the mixed norm total variation regularizations to optimize the precision of the solution. We conduct experiments on 1D linear and 2D magnetotelluric geophysical model enhancement to confirm the validity of the proposed convolution approximation theory and model enhancement algorithm. Results indicate that the proposed method significantly improves geophysical models. In particular, the 2D enhancement experiment shows that the proposed algorithm increases overall model precision by 75%. [Copyright &y& Elsevier]

Convex alternating projection algorithms (CAP) can be applied to solve multiple objective and hierarchical optimization problems in the area of control systems design involving both the ℋ2 and the ℋ∞ norms. The paper presents an application of a CAP algorithm for solving a simple problem in this area. This relies on the solution of an extremum problem in Hilbert space used to calculate a required projection onto a closed convex set. [ABSTRACT FROM PUBLISHER]

Abstract: Centroidal Voronoi tessellations and optimal Delaunay triangulations can be approximated efficiently by non-linear optimisation algorithms. This paper demonstrates that the point distribution used to initialise the optimisation algorithms is important. Compared to conventional random initialisation, certain low-discrepancy point distributions help convergence towards more spatially regular results and require fewer iterations for planar and volumetric tessellations. [Copyright &y& Elsevier]

Optimality functions define stationarity in nonlinear programming, semi-infinite optimization, and optimal control in some sense. In this paper, we consider optimality functions for stochastic programs with nonlinear, possibly nonconvex, expected value objective and constraint functions. We show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer. We construct confidence intervals for the value of the optimality function at a candidate point and, hence, provide a quantitative measure of solution quality. Based on sample average approximations, we develop an algorithm for classes of stochastic programs that include CVaR-problems and utilize optimality functions to select sample sizes. [ABSTRACT FROM AUTHOR]

We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected (every node knows at the end of the process whether H has the specified property or not). We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and s-t cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the hardness of distributed approximation for many classical optimization problems including minimum spanning tree (MST), shortest paths, and minimum cut. Many of these results are the first nontrivial lower bounds for both exact and approximate distributed computation, and they resolve previous open questions. Moreover, our unconditional lower bound of approximating MST subsumes and improves upon the previous hardness of approximation bound of Elkin [M. Elkin, SIAM J. Comput., 36 (2006), pp. 433-456] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [D. Peleg and V. Rubinovich, SIAM J. Comput., 30 (2000), pp. 1427-1442]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm for any approximation factor. Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems. [ABSTRACT FROM AUTHOR]

This paper proposes a new approximation scheme which is an extension to the well-established Charef s approximation method for fractional order systems. The method relaxes the constraints on the locations of the pole-zero pairs of the approximate system, which provides more flexibility and space for approximate system to approach the original system as close as possible. The approxi-mate system based on an optimization process performs not only a good magnitude fitting but also a good phase fitting. The benefits from using the proposed scheme are illustrated by numerical examples in frequency domains. [ABSTRACT FROM AUTHOR]

We consider approximation algorithms for nonnegative polynomial optimization over unit spheres. Such optimization models have wide applications, e.g., in signal and image processing, high order statistics, and computer vision. Since polynomial functions are nonconvex, the problems under consideration are all NP-hard. In this paper, based on convex polynomial optimization relaxations, we propose polynomial-time approximation algorithms with new approximation bounds. Numerical results are reported to show the effectiveness of the proposed approximation algorithms. [ABSTRACT FROM AUTHOR]