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With this note, we want to open a discussion on the duality theory for Interval Optimization. We begin by showing that some duality results Wu ( J Optim Theory Appl 144:615-628, ) are useless. [ABSTRACT FROM AUTHOR]

In this note, it is shown that there is error in the existing technique (Khan, I.U., Ahmad, T., Maan, N: A simplified novel technique for solving fully fuzzy linear programming problems. J. Optim. Theory Appl. 159:536-546, ). [ABSTRACT FROM AUTHOR]

This paper combines shape optimization and homogenization techniques in searching for the optimal design of microstructures in elastic scaffolds. The development of materials with specific properties is of practical interest, for example, for medical applications or for the development of lightweight structures in aeronautics. In particular, the optimal design of microstructures leads to fundamental questions for elastic porous media: how to calculate a microstructure leading to a target effective Hooke tensor. We propose a robust approach to find a design that is as insensitive as possible to domain variations. Our strategy is based on the shape derivative for the problem of achieving a prescribed effective tensor. We demonstrate the applicability and feasibility of our approach through numerical experiments. [ABSTRACT FROM AUTHOR]

This article provides instructions in preparing a paper for publication in the "Journal of Optimization Theory and Applications." Some of these guidelines are the following: 1)submission of manuscripts in triplicate; 2) reference to English as the official language of the journal; 3) inclusion of an abstract of at least 50 to 100 words in each contribution; and, 4) the abstract should be followed by a list of four to five key words identifying the subject.

Presents manuscript guidelines for papers to be published in the "Journal of Optimization Theory and Applications."

The author replies to a note on the paper 'Duality Theory for Optimization Problems with Interval-Valued Objective Function' by saying that the note is invalid. [ABSTRACT FROM AUTHOR]

This paper proposes a multi-scale method to design a continuous-time distributed algorithm for constrained convex optimization problems by using multi-agents with Markov switched network dynamics and noisy inter-agent communications. Unlike most previous work which mainly puts emphasis on dealing with fixed network topology, this paper tackles the challenging problem of investigating the joint effects of stochastic networks and the inter-agent communication noises on the distributed optimization dynamics, which has not been systemically studied in the past literature. Also, in sharp contrast to previous work in constrained optimization, we depart from the use of projected gradient flow which is non-smooth and hard to analyze; instead, we design a smooth optimization dynamics which leads to easier convergence analysis and more efficient numerical simulations. Moreover, the multi-scale method presented in this paper generalizes previously known distributed convex optimization algorithms from the fixed network topology to the switching case and the stochastic averaging obtained in this paper is a generalization of the existing deterministic averaging. [ABSTRACT FROM AUTHOR]

In deep learning, asynchronous parallel stochastic gradient descent (APSGD) is a broadly used algorithm to speed up the training process. In asynchronous system, the time delay of stale gradients in asynchronous algorithms is generally proportional to the total number of workers. When the number of workers is too large, the time delay will bring additional deviation from the accurate gradient due to delayed gradients and may cause a negative influence on the convergence speed of the algorithm. One may ask: How many workers can be used at most to achieve both the convergence and the speedup? In this paper, we consider the asynchronous training problem with the non-convex case. We theoretically study this problem to find an approximating second-order stationary point using asynchronous algorithms in non-convex optimization and investigate the behaviors of APSGD near-saddle points. This work gives the first theoretical guarantee to find an approximating second-order stationary point in asynchronous algorithms and a provable upper bound for the time delay. The techniques we provide can be generalized to analyze other types of asynchronous algorithms to understand the behaviors of asynchronous algorithms in distributed asynchronous parallel training. [ABSTRACT FROM AUTHOR]

The numerical algorithm by Cristofari et al. modifies the augmented Lagrangian method ALGENCAN proposed by Andreani and his collaborators by incorporating certain second-order information into the augmented Lagrangian framework. Professor Franco Giannessi, University of Pisa, is an outstanding mathematician whose contributions to optimization theory and its applications and to the world optimization community are difficult to overstate. The paper by Izmailov and Solodov introduces and develops a novel perturbed augmented Lagrangian method framework for constrained optimization problems. [Extracted from the article]

This paper considers an optimal control problem governed by nonlinear fractional-order systems with multiple time-varying delays and subject to canonical constraints, where the fractional-order derivatives are expressed in the Caputo sense. To solve the problem by discretization scheme, an explicit numerical integration technique is proposed for solving the fractional-order system, and the trapezoidal rule is introduced to approximate the cost functional. Then, the gradients of the resulting cost and constraint functions are derived. On the basis of this result, we develop a gradient-based optimization algorithm to numerically solve the discretized problem. Finally, numerical results of several non-trivial examples are provided to illustrate the applicability and effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The class of quaternion matrix optimization (QMO) problems, with quaternion matrices as decision variables, has been widely used in color image processing and other engineering areas in recent years. However, optimization theory for QMO is far from adequate. The main objective of this paper is to provide necessary theoretical foundations on optimality analysis, in order to enrich the contents of optimization theory and to pave way for the design of efficient numerical algorithms as well. We achieve this goal by conducting a thorough study on the first-order and second-order (sub)differentiation of real-valued functions in quaternion matrices, with a newly introduced operation called R-product as the key tool for our calculus. Combining with the classical optimization theory, we establish the first-order and the second-order optimality analysis for QMO. Particular treatments on convex functions, the ℓ 0 -norm and the rank function in quaternion matrices are tailored for a sparse low rank QMO model, arising from color image denoising, to establish its optimality conditions via stationarity. [ABSTRACT FROM AUTHOR]

In this paper, we retrieve data about the frequent users of electronic commerce during the period 2011–2016 from the Spanish National Institute of Statistics. These data, coming from surveys, have intrinsic uncertainty that we describe using appropriate random variables. Then, we propose a stochastic model to study the dynamics of frequent users of electronic commerce. The goal of this paper is to solve the inverse problem that consists of determining the model parameters as suitable parametric random variables, in such a way the model output be capable of capturing the data uncertainty, at the time instants where sample data are available, via adequate probability density functions. To achieve the aforementioned goal, we propose a computational procedure that involves building a nonlinear objective function, based on statistical moment measures, to be minimized using a variation of the particle swarm optimization algorithm. [ABSTRACT FROM AUTHOR]

The nonlinear filtering problem plays a fundamental role in multiple space-related applications. This paper offers a new filtering technique that combines Monte Carlo time propagation with a Gaussian mixture model measurement update. Differential algebra (DA) techniques are used as a tool to reduce the computational effort required by particle filters. Moreover, the use of expectation maximization (EM) optimization algorithm leads to a good approximation of the probability density functions. The performance of the new method is assessed in the nonlinear orbit determination problem, for the challenging case of low observations frequency, and in the restricted three-body dynamics. [ABSTRACT FROM AUTHOR]

A composite control law for stabilizing a small-scale helicopter during hover flight mode is proposed in this paper. The proposed method is developed by combining the sliding mode control (SMC) and a nonlinear control law. Parameters of the proposed controller are optimized using particle swarm optimization technique. The SMC ensures robustness under disturbances and parameter uncertainties, while the nonlinear control law improves the transient response of the closed-loop system. Adequacy of the combination of the sliding mode controller and the nonlinear control law is validated using mathematical analysis and simulation experiments. The results illustrate an efficient execution of the proposed controller to stabilize the system by reducing the deviations from the trim state and alleviating the effect of disturbance in the closed-loop response. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method is developed for solving a class of delay fractional optimal control problems involving nonlinear time-delay systems and subject to state inequality constraints. The fractional derivatives in this class of problems are described in the sense of Caputo, and they can be of different orders. First, we propose a numerical integration scheme for the fractional time-delay system and prove that the convergence rate of the numerical solution to the exact one is of second order based on Taylor expansion and linear interpolation. This gives rise to a discrete-time optimal control problem. Then, we derive the gradient formulas of the cost and constraint functions with respect to the decision variables and present a gradient computation procedure. On this basis, a gradient-based optimization algorithm is developed to solve the resulting discrete-time optimal control problem. Finally, several example problems are solved to demonstrate the effectiveness of the developed solution approach. [ABSTRACT FROM AUTHOR]

This paper extends the classical economic order quantity inventory model to that the planning horizon consists of two stages—a finite planning horizon and an infinite planning horizon, the demand in each stage is deterministic and stable but differs. The main goal is to find the optimal replenishment and stocking policy in each stage in order to keep the total relevant cost as low as possible, which is formulated as a mixed integer optimization problem. Using the alternating minimization method and the optimization theory, we develop a closed-form solution to the optimal inventory model and provide an optimal replenishment strategy to the retailer. Some numerical experiments are made to test the validity of the model and the effect of the involved parameters to the replenishment policy. A numerical example shows a counterintuitive fact that the economic ordering quantity may not necessarily be optimal for this inventory mechanism. [ABSTRACT FROM AUTHOR]

Nowadays, large part of the technical knowledge associated with collapses of slabs is based on past failures of bridges, floors, flat roofs and balconies. Collapse mechanisms tend to often differ from each other due to unique features which make it difficult to derive a generalised technique that can predict the right mechanism. This paper proposes a novel algorithm for tackling the problem of detection of collapse mechanisms, which is part of a pseudo-lower bound method for assessing concrete slabs in bridges and buildings. The problem is generalised to a combinatorial one, and the solution is based on a set of well-known combinatorial optimization algorithms. The proposed approach enables an identification of the domain of existence of yield-lines potentially leading to collapse. The output provides an estimation of a hampered domain of feasible yield-lines through which engineers can quickly identify zones of the slab and directions in which yield-lines leading to collapse are more likely to occur. Numerical applications of the algorithm are presented herein. [ABSTRACT FROM AUTHOR]

During the past few decades, multi-agent optimization problems have drawn increased attention from the research community. When multiple objective functions are present among agents, many works optimize the sum of these objective functions. However, this formulation implies a decision regarding the relative importance of each objective: optimizing the sum is a special case of a multi-objective problem in which all objectives are prioritized equally. To enable more general prioritizations, we present a distributed optimization algorithm that explores Pareto optimal solutions for non-homogeneously weighted sums of objective functions. This exploration is performed through a new rule based on agents' priorities that generates edge weights in agents' communication graph. These weights determine how agents update their decision variables with information received from other agents in the network. Agents initially disagree on the priorities of objective functions, though they are driven to agree upon them as they optimize. As a result, agents still reach a common solution. The network-level weight matrix is (non-doubly) stochastic, contrasting with many works on the subject in which the network-level weight matrix is doubly-stochastic. New theoretical analyses are therefore developed to ensure convergence of the proposed algorithm. This paper provides a gradient-based optimization algorithm, proof of convergence to solutions, and convergence rates of the proposed algorithm. It is shown that agents' initial priorities influence the convergence rate of the proposed algorithm and that these initial choices affect its long-run behavior. Numerical results performed with different numbers of agents illustrate the performance and effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a decentralized algorithm for non-convex optimization over tree-structured networks. We assume that each node of this network can solve small-scale optimization problems and communicate approximate value functions with its neighbors based on a novel multi-sweep communication protocol. In contrast to existing parallelizable optimization algorithms for non-convex optimization, the nodes of the network are neither synchronized nor assign any central entity. None of the nodes needs to know the whole topology of the network, but all nodes know that the network is tree-structured. We discuss conditions under which locally quadratic convergence rates can be achieved. The method is illustrated by running the decentralized asynchronous multi-sweep protocol on a radial AC power network case study. [ABSTRACT FROM AUTHOR]

Decentralized algorithms solve multi-agent problems over a connected network, where the information can only be exchanged with the accessible neighbors. Though there exist several decentralized optimization algorithms, there are still gaps in convergence conditions and rates between decentralized and centralized algorithms. In this paper, we fill some gaps by considering two decentralized algorithms: EXTRA and NIDS. They both converge linearly with strongly convex objective functions. We will answer two questions regarding them. What are the optimal upper bounds for their stepsizes? Do decentralized algorithms require more properties on the functions for linear convergence than centralized ones? More specifically, we relax the required conditions for linear convergence of both algorithms. For EXTRA, we show that the stepsize is comparable to that of centralized algorithms. For NIDS, the upper bound of the stepsize is shown to be exactly the same as the centralized ones. In addition, we relax the requirement for the objective functions and the mixing matrices. We provide the linear convergence results for both algorithms under the weakest conditions. [ABSTRACT FROM AUTHOR]

This paper focuses on a unified approach to characterizing different kinds of multiobjective robustness concepts. Based on linear and nonlinear scalarization results for several set order relations, together with the help of image space analysis, some suitable subsets of scalarization image space are introduced to make equivalent characterizations for upper set (lower set, set, certainly, respectively) less ordered robustness for uncertain multiobjective optimization problems. In particular, the nonlinear scalarization functional plays a significant role in computing various multiobjective robust solutions. Finally, the corresponding examples are included to show the effectiveness of the results derived in this paper. [ABSTRACT FROM AUTHOR]

Lower semi-continuity from above or upper semi-continuity from below has been used by many authors in recent papers. In this paper, we first study the weak semi-continuity for vector functions having particular form as that of Browder in ordered normed vector spaces; we obtain several new results on the lower semi-continuity from above or upper semi-continuity from below for these vector functions. Our results generalize some well-known results of Browder in scalar case. Secondly, we study the minimum or maximum problems for vector functions satisfying lower semi-continuous from above or upper semi-continuous from below conditions; several new results on the existence of minimal points or maximal points are obtained. We also use these results to study vector equilibrium problems and von Neumann’s minimax principle in ordered normed vector spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific application to optimization theory, we characterize the continuous representability of all continuous total preorders, which admit both a maximal and a minimal element. [ABSTRACT FROM AUTHOR]

The scalar finite-dimensional case has been discussed in the first part of this work series, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. Based on this preliminary result, the present paper dedicates itself to further study the regularity conditions for vector constrained extremum problems in a Euclidean space. The case of infinite-dimensional image will be the subject of a subsequent paper. [ABSTRACT FROM AUTHOR]

In this paper, by virtue of the image space analysis, the general scalar robust optimization problems under the strictly robust counterpart are considered, among which, the uncertainties are included in the objective as well as the constraints. Besides, on the strength of a corrected image in a new type, an equivalent relation between the uncertain optimization problem and its image problem is also established, which provides an idea to tackle with minimax problems. Furthermore, theorems of the robust weak alternative as well as sufficient characterizations of robust optimality conditions are achieved on the frame of the linear and nonlinear (regular) weak separation functions. Moreover, several necessary and sufficient optimality conditions, especially saddle point sufficient optimality conditions for scalar robust optimization problems, are obtained. Finally, a simple example for finding a shortest path is included to show the effectiveness of the results derived in this paper. [ABSTRACT FROM AUTHOR]

Following the idea of the conjecture for semi-infinite programming in a paper by Kortanek and Zhang, recently published in Optimization, in this paper we show that the Fourier-Motzkin elimination is not needed in the study of the strong duality and dual pricing properties for semi-infinite programming. We also prove several new results on the strong duality and dual pricing properties. Specifically, we propose a new subspace, under which the strong duality property holds. We give a necessary and sufficient condition for the dual pricing property to hold under this subspace, which is further used to examine the examples presented in the Basu-Martin-Ryan paper. [ABSTRACT FROM AUTHOR]

In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems. [ABSTRACT FROM AUTHOR]

This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones. [ABSTRACT FROM AUTHOR]

In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush-Kuhn-Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush-Kuhn-Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush-Kuhn-Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality. [ABSTRACT FROM AUTHOR]

This paper considers the problem of unconstrained minimization of smooth convex functions having Lipschitz continuous gradients with known Lipschitz constant. We recently proposed the optimized gradient method for this problem and showed that it has a worst-case convergence bound for the cost function decrease that is twice as small as that of Nesterov's fast gradient method, yet has a similarly efficient practical implementation. Drori showed recently that the optimized gradient method has optimal complexity for the cost function decrease over the general class of first-order methods. This optimality makes it important to study fully the convergence properties of the optimized gradient method. The previous worst-case convergence bound for the optimized gradient method was derived for only the last iterate of a secondary sequence. This paper provides an analytic convergence bound for the primary sequence generated by the optimized gradient method. We then discuss additional convergence properties of the optimized gradient method, including the interesting fact that the optimized gradient method has two types of worst-case functions: a piecewise affine-quadratic function and a quadratic function. These results help complete the theory of an optimal first-order method for smooth convex minimization. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approach to stochastic complementarity problems which allows to take into account various notions of risk aversion. Our model enhances the expected residual minimization formulation of Chen and Fukushima (Math Oper Res 30:1022-1038, 2005) by replacing the expectation with a more general convex, nondecreasing and law-invariant risk functional. Relevant examples of such risk functionals include the Expected Excess and the Conditional Value-at-Risk. We examine qualitative stability of the resulting one-stage stochastic optimization problem with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the decision vector and the entering probability measure. By a classical result from parametric optimization, this implies upper semicontinuity of the optimal value function. Throughout the analysis, we assume for building the model that a nonlinear complementarity function fulfills a certain polynomial growth condition. We conclude the paper demonstrating that this assumption holds in the vast majority of all practically relevant cases. [ABSTRACT FROM AUTHOR]

This paper combines variable ordering structures with set relations in set optimization. We continue our examinations of set optimization problems and use the optimality notions as well as the set relations introduced in the paper (Eichfelder and Pilecka in Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach, 2016). We present linear scalarization results and a new nonlinear scalarization approach for set relations. Moreover, we introduce some additional set relations which allow us to investigate connections between our results, based on the set approach, and others presented in the literature which are based on the vector approach. This gives us the possibility to apply the optimality conditions derived there to our concepts. [ABSTRACT FROM AUTHOR]

In this paper, problems of mathematical diagnostics are considered. The most popular approach to these problems is based on statistical methods. In this paper, the author treats the mentioned problems by means of optimization. This approach can be useful in the case where statistical characteristics of the database are unknown or the database is not sufficiently large. In this paper, a nonsmooth model is used where it is required to separate two sets, whose convex hulls may intersect. A linear classifier is used to identify the points of two sets. The quality of identification is evaluated by the so-called natural functional, based on the number of misclassified points. It is required to find the optimal hyperplane, which minimizes the number of misclassified points by means of the translation and rotation operations. Since the natural functional (number of misclassified points) is discontinuous, it is suggested to approximate it by some surrogate functional possessing at least the continuity property. In this paper, two surrogate functionals are introduced and studied. It is shown that one of them is subdifferentiable, and the second one is continuously differentiable. It is also demonstrated that the theory of exact penalization can be employed to reduce the given constrained optimization problems to an unconstrained one. Numerical methods are constructed, where the steepest descent directions of the surrogate functionals are used to minimize the natural one. Necessary conditions for a minimum are formulated for both surrogate functionals. [ABSTRACT FROM AUTHOR]

In this paper, we present a primal-dual interior point method for linear optimization problems based on a new efficient kernel function with a trigonometric barrier term. We derive the complexity bounds for large and small-update methods, respectively. We obtain the best known complexity bound for large update, which improves significantly the so far obtained complexity results based on a trigonometric kernel function given by Peyghami et al. The results obtained in this paper are the first to reach this goal. [ABSTRACT FROM AUTHOR]

This paper studies opinion dynamics for a set of heterogeneous populations of individuals pursuing two conflicting goals: to seek consensus and to be coherent with their initial opinions. The multi-population game under investigation is characterized by (i) rational agents who behave strategically, (ii) heterogeneous populations, and (iii) opinions evolving in response to local interactions. The main contribution of this paper is to encompass all of these aspects under the unified framework of mean-field game theory. We show that, assuming initial Gaussian density functions and affine control policies, the Fokker-Planck-Kolmogorov equation preserves Gaussianity over time. This fact is then used to explicitly derive expressions for the optimal control strategies when the players are myopic. We then explore consensus formation depending on the stubbornness of the involved populations: We identify conditions that lead to some elementary patterns, such as consensus, polarization, or plurality of opinions. Finally, under the baseline example of the presence of a stubborn population and a most gregarious one, we study the behavior of the model with a finite number of players, describing the dynamics of the average opinion, which is now a stochastic process. We also provide numerical simulations to show how the parameters impact the equilibrium formation. [ABSTRACT FROM AUTHOR]

As a complement to a recent paper by An and Yen (Appl Anal 94:108-128, ) on subdifferentials of the optimal value function in parametric convex programming under inclusion constraints and functional constraints, this paper studies the differential stability of convex optimization problems under a regularity condition of Aubin's type (Aubin in Optima and equilibria: an introduction to nonlinear analysis. Springer, New York, ). By a suitable sum rule for convex subdifferentials, we obtain exact formulas for the subdifferential and singular subdifferential of the optimal value function. Illustrative examples and a detailed comparison of our results with those of the above-mentioned paper are given. [ABSTRACT FROM AUTHOR]

In this paper, we provide an algorithm for solving constrained composite primal-dual monotone inclusions, i.e., monotone inclusions in which a priori information on primal-dual solutions is represented via closed and convex sets. The proposed algorithm incorporates a projection step onto the a priori information sets and generalizes methods proposed in the literature for solving monotone inclusions. Moreover, under the presence of strong monotonicity, we derive an accelerated scheme inspired on the primal-dual algorithm applied to the more general context of constrained monotone inclusions. In the particular case of convex optimization, our algorithm generalizes several primal-dual optimization methods by allowing a priori information on solutions. In addition, we provide an accelerated scheme under strong convexity. An application of our approach with a priori information is constrained convex optimization problems, in which available primal-dual methods impose constraints via Lagrange multiplier updates, usually leading to slow algorithms with unfeasible primal iterates. The proposed modification forces primal iterates to satisfy a selection of constraints onto which we can project, obtaining a faster method as numerical examples exhibit. The obtained results extend and improve several results in the literature. [ABSTRACT FROM AUTHOR]

Quadratically constrained quadratic programming arises from a broad range of applications and is known to be among the hardest optimization problems. In recent years, semidefinite relaxation has become a popular approach for quadratically constrained quadratic programming, and many results have been reported in the literature. In this paper, we first discuss how to assess the gap between quadratically constrained quadratic programming and its semidefinite relaxation. Based on the estimated gap, we discuss how to construct an exact penalty function for quadratically constrained quadratic programming based on its semidefinite relaxation. We then introduce a special penalty method for quadratically constrained linear programming based on its semidefinite relaxation, resulting in the so-called conditionally quasi-convex relaxation. We show that the conditionally quasi-convex relaxation can provide tighter bounds than the standard semidefinite relaxation. By exploring various properties of the conditionally quasi-convex relaxation model, we develop two effective procedures, an iterative procedure and a bisection procedure, to solve the constructed conditionally quasi-convex relaxation. Promising numerical results are reported. [ABSTRACT FROM AUTHOR]

In this paper, we propose the descent method with new inexact line-search for unconstrained optimization problems on Riemannian manifolds. The global convergence of the proposed method is established under some appropriate assumptions. We further analyze some convergence rates, namely R-linear convergence rate, superlinear convergence rate and quadratic convergence rate, of the proposed descent method. [ABSTRACT FROM AUTHOR]

The Barzilai-Borwein conjugate gradient methods, which were first proposed by Dai and Kou (Sci China Math 59(8):1511-1524, 2016), are very interesting and very efficient for strictly convex quadratic minimization. In this paper, we present an efficient Barzilai-Borwein conjugate gradient method for unconstrained optimization. Motivated by the Barzilai-Borwein method and the linear conjugate gradient method, we derive a new search direction satisfying the sufficient descent condition based on a quadratic model in a two-dimensional subspace, and design a new strategy for the choice of initial stepsize. A generalized Wolfe line search is also proposed, which is nonmonotone and can avoid a numerical drawback of the original Wolfe line search. Under mild conditions, we establish the global convergence and the R-linear convergence of the proposed method. In particular, we also analyze the convergence for convex functions. Numerical results show that, for the CUTEr library and the test problem collection given by Andrei, the proposed method is superior to two famous conjugate gradient methods, which were proposed by Dai and Kou (SIAM J Optim 23(1):296-320, 2013) and Hager and Zhang (SIAM J Optim 16(1):170-192, 2005), respectively. [ABSTRACT FROM AUTHOR]

This paper presents a numerical algorithm for solving a class of nonlinear optimal control problems subject to a system of fractional differential equations. We first propose a robust second-order numerical integration scheme for the system. The objective is approximated by the trapezoidal rule. We then apply a gradient-based optimization method to the discretized problem. Formulas for calculating the gradients are derived. Computational results demonstrate that our method is able to generate accurate numerical approximations for problems with multiple states and controls. It is also robust with respect to the fractional orders of derivatives. [ABSTRACT FROM AUTHOR]

We propose a heuristic optimization method for a density-based fluid topology optimization using a Hessian matrix. In flow topology optimization, many researches use a gradient-based method. Convergence rate of a gradient method is linear, which means slow convergence near the optimal solution. For faster convergence, we utilize a Hessian matrix toward the end of the optimization procedure. In the present paper, we formulate a fluid optimization problem using the lattice Boltzmann method and heuristically solve the optimization problem with using an approximated sensitivity. In the formulation of a Hessian matrix, we use a heuristic trick in order to formulate it as a diagonal matrix. By the heuristics, the computation cost is decreased drastically. The validity of the method is studied via numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems. [ABSTRACT FROM AUTHOR]

This paper aims at providing some formulae for the subdifferential and the conjungate function of the supremum function over an arbitrary family of functions. The work is principally motivated by the case when data functions are lower semicontinuous proper and convex. Nevertheless, we explore the case when the family of functions is arbitrary, but satisfying that the biconjugate of the supremum functions is equal to the supremum of the biconjugate of the data functions. The study focuses its attention on functions defined in finite-dimensional spaces; in this case, the formulae can be simplified under certain qualification conditions. However, we show how to extend these results to arbitrary locally convex spaces, without any qualification condition. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic approximation algorithm for optimization of functions based on an adaptive extremum seeking method. The essence of this method is to approximate the gradient direction by introduction of a probing sequence, that is added to approximations and subsequently demodulated using an adaptive gain. Assuming that the probing and the demodulation signals are martingale difference sequences with adaptive diminishing gains, it is proved that the approximations converge almost surely to the optimizing value, under mild constraints on the measurement disturbance, and without assuming a priori boundedness of the approximation sequence. The measurement disturbance can contain a stochastic component, as well as a mean-square bounded deterministic component. The stochastic component can be nonstationary colored noise or a state-dependent random sequence. [ABSTRACT FROM AUTHOR]

In this paper, we present the Wolfe’s reduced gradient method for multiobjective (multicriteria) optimization. We precisely deal with the problem of minimizing nonlinear objectives under linear constraints and propose a reduced Jacobian method, namely a reduced gradient-like method that does not scalarize those programs. As long as there are nondominated solutions, the principle is to determine a direction that decreases all goals at the same time to achieve one of them. Following the reduction strategy, only a reduced search direction is to be found. We show that this latter can be obtained by solving a simple differentiable and convex program at each iteration. Moreover, this method is conceived to recover both the discontinuous and continuous schemes of Wolfe for the single-objective programs. The resulting algorithm is proved to be (globally) convergent to a Pareto KKT-stationary (Pareto critical) point under classical hypotheses and a multiobjective Armijo line search condition. Finally, experiment results over test problems show a net performance of the proposed algorithm and its superiority against a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort. [ABSTRACT FROM AUTHOR]

Modern radars are highly flexible systems, relying on digital antennas to dynamically control the radar beam shape and position through electronic circuits. Radar surveillance is performed by sequential emission of different radar beams. Optimization of radar surveillance requires finding a minimal subset of radar beams, which covers and ensures detection over the surveillance space, among a collection of available radar beams with different shapes and positions, thus minimizing the required scanning time. Optimal radar surveillance can be modelled by grid covering, a specific geometric case of set covering where the universe set is laid out on a grid, representing the radar surveillance space, which must be covered using available subsets, representing the radar beams detection areas. While the set cover problem is generally difficult to solve optimally, certain geometric cases can be optimized in polynomial time. This paper studies the theoretical complexity of grid cover problems used for modelling radar surveillance, proving that unidimensional grids can be covered by strongly polynomial algorithms based on dynamic programming, whereas optimal covering of bidimensional grids is generally non-deterministic polynomially hard. [ABSTRACT FROM AUTHOR]

In this paper, we establish two intersection theorems which are useful in considering some optimization problems (complementarity problems, variational inequalities, minimax inequalities, saddle point problems). [ABSTRACT FROM AUTHOR]

In this paper, we incorporate importance sampling strategy into accelerated framework of stochastic alternating direction method of multipliers for solving a class of stochastic composite problems with linear equality constraint. The rates of convergence for primal residual and feasibility violation are established. Moreover, the estimation of variance of stochastic gradient is improved due to the use of important sampling. The proposed algorithm is capable of dealing with the situation, where the feasible set is unbounded. The experimental results indicate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, robust efficient solutions of a vector optimization problem, whose image space is ordered by an arbitrary ordering cone, are defined. This is done from different points of view, including set based and norm based. The relationships between these solution concepts are established. Furthermore, it is shown that, for a general vector optimization problem, each norm-based robust efficient solution is a strictly efficient solution; each isolated efficient solution is a norm-based robust efficient solution; and, under appropriate assumptions, each norm-based robust efficient solution is a Henig properly efficient solution. Various necessary and sufficient conditions for characterizing norm-based robust solutions of a general vector optimization problem, in terms of the tangent and normal cones and the nonascent directions, are presented. An optimization problem for calculating a robustness radius is provided, and then, the largest robustness radius is determined. Moreover, some alterations of objective functions preserving weak/strict/Henig proper/robust efficiency are studied. [ABSTRACT FROM AUTHOR]