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We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the mapwhere it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology--that is, theweakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional--if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C8 maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C8 maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green's theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction. [ABSTRACT FROM AUTHOR]

We propose an indefinite proximal subgradient-based algorithm (IPSB) for solving nonsmooth composite optimization problems. IPSB is a generalization of the Nesterov's dual algorithm, where an indefinite proximal term is added to the subproblems, which can make the subproblem easier and the algorithm efficient when an appropriate proximal operator is judiciously setting down. Under mild assumptions, we establish sublinear convergence of IPSB to a region of the optimal value. We also report some numerical results, demonstrating the efficiency of IPSB in comparing with the classical dual averaging-type algorithms. [ABSTRACT FROM AUTHOR]

We investigate a solution of a convex programming problem with a strongly convex objective function based on the dual approach. A dual optimization problem has constraints on the positivity of variables. We study the methods and properties of transformations of dual variables that enable us to obtain an unconstrained optimization problem. We investigate the previously known method of transforming the components of dual variables in the form of their modulus (modulus method). We show that in the case of using the modulus method, the degree of the degeneracy of the function increases as it approaches the optimal point. Taking into account the ambiguity of the gradient in the boundary regions of the sign change of the new dual function variables and the increase in the degree of the function degeneracy, we need to use relaxation subgradient methods (RSM) that are difficult to implement and that can solve non-smooth non-convex optimization problems with a high degree of elongation of level surfaces. We propose to use the transformation of the components of dual variables in the form of their square (quadratic method). We prove that the transformed dual function has a Lipschitz gradient with a quadratic method of transformation. This enables us to use efficient gradient methods to find the extremum. The above properties are confirmed by a computational experiment. With a quadratic transformation compared to a modulus transformation, it is possible to obtain a solution of the problem by relaxation subgradient methods and smooth function minimization methods (conjugate gradient method and quasi-Newtonian method) with higher accuracy and lower computational costs. The noted transformations of dual variables were used in the program module for calculating the maximum permissible emissions of enterprises (MPE) of the software package for environmental monitoring of atmospheric air (ERA-AIR). [ABSTRACT FROM AUTHOR]

I construct a new cutting-plane model for approximating nonsmooth nonconvex functions in multiobjective optimization and propose a new bundle-type method with the help of an improvement function. The presented bundle method possesses three features. Firstly, the objective and constraint functions are approximated by a new cutting-plane model, which is a local convexification of the corresponding functions, instead of the entire approximation for the functions, as most bundle methods do. Secondly, the subgradients and values of the objective and constraint functions are computed approximately. In other words, approximate calculation is applied to the method, and the proposed algorithm is doubly approximate to some extent. Thirdly, the introduction of the improvement function eliminates the necessity of employing any scalarization, which is the usual method when dealing with multiobjective optimization. Under reasonable conditions satisfactory convergence results are obtained.

In this paper, we introduce a new algorithm for finding a solution of an equilibrium problem in a real Hilbert space. Our paper extends the single projection method to pseudomonotone variational inequalities, from a 2018 paper of Shehu et. al., to pseudomonotone equilibrium problems in a real Hilbert space. On the basis of the given algorithm for the equilibrium problem, we develop a new algorithm for finding a common solution of a equilibrium problem and fixed point problem. The strong convergence of the algorithm is established under mild assumptions. Several of fundamental experiments in finite (infinite) spaces are provided to illustrate the numerical behavior of the algorithm for the equilibrium problem and to compare it with other algorithms. [ABSTRACT FROM AUTHOR]

This paper studies the problem of energy efficiency (EE) maximization via user association, power, and backhaul (BH) flow control in the downlink of millimeter wave BH heterogeneous networks. This problem is mathematically formulated as a mixed-integer non-linear program, which is non-convex. To get a tractable solution, the initial problem is separated into two sub-problems and optimized sequentially. The first is a joint user association and power control sub-problem for the access network (AN) (AN sub-problem). The second is a joint flow and power control sub-problem for the BH network (BH sub-problem). While the BH sub-problem is a convex optimization problem and hence can be efficiently solved, the AN sub-problem assumes the form of a generalized assignment problem, which is known to be NP-hard. To that end, we utilize Lagrangian decomposition to propose two polynomial time solution techniques that obtain a high-quality solution for the AN sub-problem. The first, referred to as Technique A, uses dynamic programming, the subgradient method, and a heuristic. The second, named Technique B, uses the multiplier adjustment method, the sorting algorithm, and a heuristic. Simulation results are used to demonstrate the effectiveness of the proposed energy efficient user association, power, and BH flow control algorithms as compared with benchmark user association schemes that incorporate the BH sub-problem algorithm, in terms of the total AN power, BH power, and overall network (AN plus BH) EE. The computational complexity and practical implementation of the proposed algorithms are discussed.

We investigate a solution of a convex programming problem with a strongly convex objective function based on the dual approach. A dual optimization problem has constraints on the positivity of variables. We study the methods and properties of transformations of dual variables that enable us to obtain an unconstrained optimization problem. We investigate the previously known method of transforming the components of dual variables in the form of their modulus (modulus method). We show that in the case of using the modulus method, the degree of the degeneracy of the function increases as it approaches the optimal point. Taking into account the ambiguity of the gradient in the boundary regions of the sign change of the new dual function variables and the increase in the degree of the function degeneracy, we need to use relaxation subgradient methods (RSM) that are difficult to implement and that can solve non-smooth non-convex optimization problems with a high degree of elongation of level surfaces. We propose to use the transformation of the components of dual variables in the form of their square (quadratic method). We prove that the transformed dual function has a Lipschitz gradient with a quadratic method of transformation. This enables us to use efficient gradient methods to find the extremum. The above properties are confirmed by a computational experiment. With a quadratic transformation compared to a modulus transformation, it is possible to obtain a solution of the problem by relaxation subgradient methods and smooth function minimization methods (conjugate gradient method and quasi-Newtonian method) with higher accuracy and lower computational costs. The noted transformations of dual variables were used in the program module for calculating the maximum permissible emissions of enterprises (MPE) of the software package for environmental monitoring of atmospheric air (ERA-AIR).

The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, nonsmooth penalty formulations do not suffer from the same type of ill-conditioning. Consequently, standard algorithms for nonsmooth optimization, such as subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. Moreover, nonsmooth formulations are naturally robust against outliers. Our framework subsumes such important computational tasks as phase retrieval, blind deconvolution, quadratic sensing, matrix completion, and robust PCA. Numerical experiments on these problems illustrate the benefits of the proposed approach. [ABSTRACT FROM AUTHOR]

In the article we have obtained some estimates of the rate of convergence for the recently proposed by Yu. E.Nesterov method of minimization of a convex Lipschitz-continuous function of two variables on a square with a fixed side. The idea of the method is to divide the square into smaller parts and gradually remove them so that in the remaining sufficiently small part. The method consists in solving auxiliary problems of one-dimensional minimization along the separating segments and does not imply the calculation of the exact value of the gradient of the objective functional. The main result of the paper is proved in the class of smooth convex functions having a Lipschitz-continuous gradient. Moreover, it is noted that the property of Lipschitzcontinuity for gradient is sufficient to require not on the whole square, but only on some segments. It is shown that the method can work in the presence of errors in solving auxiliary one-dimensional problems, as well as in calculating the direction of gradients. Also we describe the situation when it is possible to neglect or reduce the time spent on solving auxiliary one-dimensional problems. For some examples, experiments have demonstrated that the method can work effectively on some classes of non-smooth functions. In this case, an example of a simple non-smooth function is constructed, for which, if the subgradient is chosen incorrectly, even if the auxiliary one-dimensional problem is exactly solved, the convergence property of the method may not hold. Experiments have shown that the method under consideration can achieve the desired accuracy of solving the problem in less time than the other methods (gradient descent and ellipsoid method) considered. Partially, it is noted that with an increase in the accuracy of the desired solution, the operating time for the Yu. E. Nesterovs method can grow slower than the time of the ellipsoid method.

This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces limit points that are all first-order stationary. More generally, our result applies to any function with a Whitney stratifiable graph. In particular, this work endows the stochastic subgradient method, and its proximal extension, with rigorous convergence guarantees for a wide class of problems arising in data science—including all popular deep learning architectures. [ABSTRACT FROM AUTHOR]

We present a subgradient method of minimization, similar to the method of minimal iterations for solving systems of equations, which inherits from the latter convergence properties on quadratic functions. The proposed algorithm, for a certain set of parameters, coincides with the previously known method of minimizing piecewise linear functions and is an element of the family of minimization methods with relaxation of the distance to extremum, developed by B.T. Polyak, where the step length is calculated based on the predefined minimum value of the function. We link parameters of this method to the constraint on the degree of homogeneity of the function and obtain estimates on its convergence rate on convex functions. We prove that on some classes of functions it converges at the rate of a geometric progression. We also discuss the computational capabilities of this approach for solving problems with high dimension. [ABSTRACT FROM AUTHOR]

Studying Bregman distance iterative methods for solving optimization problems has become an important and very interesting topic because of the numerous applications of the Bregman distance techniques. These applications are based on the type of convex functions associated with the Bregman distance. In this paper, two different extragraident methods were proposed for studying pseudomonotone variational inequality problems using Bregman distance in real Hilbert spaces. The first algorithm uses a fixed stepsize which depends on a prior estimate of the Lipschitz constant of the cost operator. The second algorithm uses a self-adaptive stepsize which does not require prior estimate of the Lipschitz constant of the cost operator. Some convergence results were proved for approximating the solutions of pseudomonotone variational inequality problem under standard assumptions. Moreso, some numerical experiments were also given to illustrate the performance of the proposed algorithms using different convex functions such as the Shannon entropy and the Burg entropy. In addition, an application of the result to a signal processing problem is also presented.

The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature.

The subgradient step selection method based on the known minimal value of function is studied in the paper. The authors show that it is an analogue of the method of minimal errors for solving linear equation systems. For a sequence of the minimum function values on the current set of method iterations, the estimate for the rate of convergence in the form of inequality is received.

This paper continues our recent effort in applying continuous optimization techniques to study optimal multicast communication networks modeled as bilevel hierarchical clustering problems. Given a finite number of nodes, we consider two different models of multicast networks by identifying a certain number of nodes as cluster centers, and at the same time, locating a particular node that serves as a total center so as to minimize the total transportation cost throughout the network. The fact that the cluster centers and the total center have to be among the given nodes makes these problems discrete optimization problems. Our approach is to reformulate the discrete problems as continuous ones and to apply Nesterov’s smoothing approximation techniques on the Minkowski gauges that are used as distance measures. This approach enables us to propose two implementable DCA-based algorithms for solving the problems. Numerical results and practical applications are provided to illustrate our approach. [ABSTRACT FROM AUTHOR]

Based on the redistributed technique of bundle methods and the auxiliary problem principle, we present a redistributed bundle method for solving a generalized variational inequality problem which consists of finding a zero point of the sum of two multivalued operators. The considered problem involves a nonsmooth nonconvex function which is difficult to approximate by workable functions. By imitating the properties of lower- C 2 functions, we consider approximating the local convexification of the nonconvex function, and the local convexification parameter is modified dynamically in order to make the augmented function produce nonnegative linearization errors. The convergence of the proposed algorithm is discussed when the sequence of stepsizes converges to zero, any weak limit point of the sequence of serious steps x k is a solution of problem (P) under some conditions. The presented method is the generalization of the convex bundle method [Salmon, G, JJ Strodiot and VH Nguyen (2004). A bundle method for solving variational inequalities. SIAM Journal on Optimization, 14(3), 869–893]. [ABSTRACT FROM AUTHOR]

In this paper we consider the first order evolutionary inclusions with nonlinear weakly continuous operators and a multivalued term which involves the Clarke subgradient of a locally Lipschitz function. First, we provide a surjectivity result for stationary inclusion with weakly–weakly upper semicontinuous multifunction. Then, we use this result to prove the existence of solutions to the Rothe sequence and the evolutionary subgradient inclusion. Finally, we apply our results to the non-stationary Navier–Stokes equation with nonmonotone and multivalued frictional boundary conditions. [ABSTRACT FROM AUTHOR]

An information technology for histogram analysis and a method for noise immunity binary processing of integrated and printed circuits board photo-masks image based on this technology was carryed out. This method based on the sub-gradient iterative noise stability optimization method satisfies the demands of the automated optical control of photo-masks and printed circuits in the error and noise immunity. The maximum binary processing error does not exceed 1 pixel if the signal-to-noise ratio is more than 7.

To discuss the existence and uniqueness of proper scoring rules one needs to extend the associated entropy functions as sublinear functions to the conic hull of the prediction set. In some natural function spaces, such as the Lebesgue Lp-spaces over Rd, the positive cones have empty interior. Entropy functions defined on such cones have directional derivatives only, which typically exist on large subspaces and behave similarly to gradients. Certain entropies may be further extended continuously to open cones in normed spaces containing signed densities. The extended densities are Gâteaux differentiable except on a negligible set and have everywhere continuous subgradients due to the supporting hyperplane theorem. We introduce the necessary framework from analysis and algebra that allows us to give an affrmative answer to the titular question of the paper. As a result of this, we give a formal sense in which entropy functions have uniquely associated proper scoring rules. We illustrate our framework by studying the derivatives and subgradients of the following three prototypical entropies: Shannon entropy, Hyvarinen entropy, and quadratic entropy. [ABSTRACT FROM AUTHOR]

Nonsmooth convex optimization problems are solved over fixed point sets of nonexpansive mappings by using a distributed optimization technique. This is done for a networked system with an operator, who manages the system, and a finite number of users, by solving the problem of minimizing the sum of the operator's and users' nondifferentiable, convex objective functions over the intersection of the operator's and users' convex constraint sets in a real Hilbert space. We assume that each of their constraint sets can be expressed as the fixed point set of an implementable nonexpansive mapping. This setting allows us to discuss nonsmooth convex optimization problems in which the metric projection onto the constraint set cannot be calculated explicitly. We propose a parallel subgradient algorithm for solving the problem by using the operator's attribution such that it can communicate with all users. The proposed algorithm does not use any proximity operators, in contrast to conventional parallel algorithms for nonsmooth convex optimization. We first study its convergence property for a constant step-size rule. The analysis indicates that the proposed algorithm with a small constant step size approximates a solution to the problem. We next consider the case of a diminishing step-size sequence and prove that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. We also give numerical examples to support the convergence analyses. [ABSTRACT FROM AUTHOR]

This paper deals with a variant relaxed CQ algorithm by using a new searching direction, which is not the gradient of a corresponding function. The strategy is to intend to improve the convergence. Its convergence is proved under some suitable conditions. Numerical results illustrate that our variant relaxed CQ algorithm converges more quickly than the existing algorithms. [ABSTRACT FROM AUTHOR]

This paper addresses the short-term generation planning (STGP) through thermoelectric units. The mathematical model is presented as a Mixed Integer Non Linear Problem (MINLP). Several works on the state of art of the problem have revealed that the computational effort of this problem grows exponentially with the number of time periods and number of thermoelectric units. Therefore, we present two alternatives to solve a STGP based on Benders' partitioning algorithm and Lagrangian relaxation in order to reduce the computational effort. The proposal is to apply primal and dual decomposition techniques, which exploit the structure of the problem to reduce solution time by decomposing the STGP into a master problem and a subproblem. For Benders' algorithm, the master problem is a Mixed Integer Problem (MIP) and for the subproblem, it is a Non Linear Problem (NLP). For Lagrangian relaxation, the master problem and the subproblem are MINLP. The computational experiments show the performance of both decomposition techniques applied to the STGP. These techniques allow us to save computation time when compared to some high performance commercial solvers. [ABSTRACT FROM AUTHOR]

This paper deals with a variant relaxed CQ algorithm by using a new searching direction, which is not the gradient of a corresponding function. The strategy is to intend to improve the convergence. Its convergence is proved under some suitable conditions. Numerical results illustrate that our variant relaxed CQ algorithm converges more quickly than the existing algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we present a simultaneous subgradient algorithm for solving the multiple-sets split feasibility problem. The algorithm employs two extrapolated factors in each iteration, which not only improves feasibility by eliminating the need to compute the Lipschitz constant, but also enhances flexibility due to applying variable step size. The convergence of the algorithm is proved under suitable conditions. Numerical results illustrate that the new algorithm has better convergence than the existing one. [ABSTRACT FROM AUTHOR]

Abstract: We consider a simplified model of methane hydrates which we cast as a nonlinear evolution problem. For its well-posedness we extend the existing theory to cover the case in which the problem involves a measurable family of graphs. We represent the nonlinearity as a subgradient and prove a useful comparison principle, thus optimal regularity results follow. For the numerical solution we apply a fully implicit scheme without regularization and use the semismooth Newton algorithm for a solver, and the graph is realized as a complementarity constraint (CC). The algorithm is very robust and we extend it to define an easy and superlinearly convergent fully implicit scheme for the Stefan problem and other multivalued examples. [Copyright &y& Elsevier]

Abstract: We show that for every compact domain in a Euclidean space with d.c. (delta-convex) boundary there exists a unique Legendrian cycle such that the associated curvature measures fulfill a local version of the Gauss–Bonnet formula. This was known in dimensions two and three and was open in higher dimensions. In fact, we show this property for a larger class of sets including also lower-dimensional sets. We also describe the local index function of the Legendrian cycles and we show that the associated curvature measures fulfill the Crofton formula. [Copyright &y& Elsevier]

This paper considers the quadratic Gaussian multiterminal (MT) source coding problem and provides a new sufficient condition for the Berger–Tung (BT) sum-rate bound to be tight. The converse proof utilizes a set of virtual remote sources given which the observed sources are block independent with a maximum block size of 2. The given MT source coding problem is then related to a set of two-terminal problems with matrix-distortion constraints, for which a new lower bound on the sum-rate is given. By formulating a convex optimization problem over all distortion matrices, a sufficient condition is derived for the optimal BT scheme to satisfy the subgradient-based Karush–Kuhn–Tucker condition. The subset of the quadratic Gaussian MT problem satisfying our new sufficient condition subsumes all previously known tight cases, and our proof technique opens a new direction for more general partial solutions. [ABSTRACT FROM AUTHOR]

Abstract: Consider the stability problem for the following linear switched system (differential inclusion) . Here   are dimensional Hurwitz stable real matrices. In this study for this system we investigate the problem of the existence and construction of a common diagonal Lyapunov function of the form where is a positive diagonal matrix. In the case of , i.e. third order system, we suggest a simple elimination algorithm which gives a common in the case of existence. [Copyright &y& Elsevier]

We extend the well-known BFGS quasi-Newton method and its memory-limited variant LBFGS to the optimization of nonsmooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: the local quadratic model, the identification of a descent direction, and the Wolfe line search conditions. We prove that under some technical conditions, the resulting subBFGS algorithm is globally convergent in objective function value. We apply its memory-limited variant (subLBFGS) to L2-regularized risk minimization with the binary hinge loss. To extend our algorithm to the multiclass and multilabel settings, we develop a new, efficient, exact line search algorithm. We prove its worst-case time complexity bounds, and show that our line search can also be used to extend a recently developed bundle method to the multiclass and multilabel settings. We also apply the direction-finding component of our algorithm to L1-regularized risk minimization with logistic loss. In all these contexts our methods perform comparable to or better than specialized state-of-the-art solvers on a number of publicly available data sets. An open source implementation of our algorithms is freely available. [ABSTRACT FROM AUTHOR]

A smoothing method of multipliers which is a natural result of cross-entropic regularization for min-max problems is analyzed. As a smoothing technique, we first show how the smooth approximation yields the first order information on the behavior of max function. Then under suitable assumptions, some basic properties including the Hessian are given. At last, the condition number is analyzed, and the results reveal that the smoothing method of multipliers is stable for any fixed smoothing parameter. [ABSTRACT FROM AUTHOR]

Several sparseness penalties have been suggested for delivery of good predictive performance in automatic variable selection within the framework of regularization. All assume that the true model is sparse. We propose a penalty, a convex combination of the L1- and L∞-norms, that adapts to a variety of situations including sparseness and nonsparseness, grouping and nongrouping. The proposed penalty performs grouping and adaptive regularization. In addition, we introduce a novel homotopy algorithm utilizing subgradients for developing regularization solution surfaces involving multiple regularizers. This permits efficient computation and adaptive tuning. Numerical experiments are conducted using simulation. In simulated and real examples, the proposed penalty compares well against popular alternatives. [ABSTRACT FROM PUBLISHER]

A novel method to reconstruct object boundaries with geodesic circular arc is proposed in this paper. Within this framework, an energy of circular arc spline is utilized to simultaneously arrange and interpolate each member in the set of sparse unorganized feature points from the desired boundaries. A general form for a family of parametric circular arc spline is firstly derived and followed by a novel method of arranging these feature points by minimizing an energy term depending on the circular arc spline configuration defined on these feature points. With regard to the fact that the energy function is usually nonconvex and nondifferentiable at its critical points, an improved scheme of particle swarm optimizer is given to find the minimum for the energy in this paper. With this improved scheme, each pair of neighboring feature points along the boundaries of the desired objects are picked out from the set of sparse unorganized feature points, and the corresponding directional chord tangent angles are computed simultaneously to finish interpolation. We show experimentally and comparatively that the proposed method can perform effectively to restrict leakage on weak boundaries and premature convergence on long concave boundaries. Besides, it has good noise robustness and can as well extract multiple and open boundaries. [ABSTRACT FROM AUTHOR]

Traditional integrated modeling (IM) is based on developing and aggregating all relevant (sub)models and data into a single integrated linear programming (LP) model. Unfortunately, this approach is not applicable for IM under asymmetric information (ASI), i.e., when "private" information regarding sectoral/regional models is not available, or it cannot be shared by modeling teams (sectoral agencies). The lack of common information about LP submodels makes LP methods inapplicable for integrated LP modeling. The aim of this paper is to develop a new approach to link and optimize distributed sectoral/regional optimization models, providing a means of decentralized cross-sectoral coordination in the situation of ASI. Thus, the linkage methodology enables the investigation of policies in interdependent systems in a "decentralized" fashion. For linkage, the sectoral/regional models do not need recoding or reprogramming. They also do not require additional data harmonization tasks. Instead, they solve their LP submodels independently and in parallel by a specific iterative subgradient algorithm for nonsmooth optimization. The submodels continue to be the same separate LP models. A social planner (regulatory agency) only needs to adjust the joint resource constraints to simple subgradient changes calculated by the algorithm. The approach enables more stable and resilient systems' performance and resource allocation as compared to the independent policies designed by separate models without accounting for interdependencies. The paper illustrates the application of the methodology to link detailed energy and agricultural production planning models under joint constraints on water and land use. [ABSTRACT FROM AUTHOR]

In this note we give some characterizations for the differentiability with respect to a bornology of a continuous convex function. The special case of seminorms is treated. A characterization of this type of differentiability in terms of the subgradient of the function is also obtained.

In this paper, we apply the primal-dual decomposition and subgradient projection methods to solve the rate-distortion optimization problem with the constant bit rate constraint. The primal decomposition method enables spatial or temporal prediction dependency within a group of picture (GOP) to be processed in the master primal problem. As a result, we can apply the dual decomposition to minimize independently the Lagrangian cost of all the MBs using the reference software model of H.264. Further- more, the optimal Lagrange multiplier λ* is iteratively derived from the solution of the dual problem. As an example, we derive the optimal bit allocation condition with the consideration of temporal prediction dependency among the pictures. Experimental results show that the proposed method achieves better performance than the reference software model of 11.264 with rate control. [ABSTRACT FROM AUTHOR]

Recently an instantaneous blind source separation (BSS) approach that exploits the bounded magnitude structure of digital communications signals has been introduced. In this paper, we introduce a deflationary adaptive algorithm based on this criterion and provide its convergence analysis. We show that the resulting algorithm is convergent to one of the globally optimal points that correspond to perfect separation. The simulation examples related to the separation of digital communication signals are provided to illustrate the convergence and the performance of the algorithm [ABSTRACT FROM PUBLISHER]

Abstract: Facility location–allocation (FLA) problem is a very important subject in today''s business. It is an important part of a company''s global logistic system. Various FLA problems have been considered in operations research (OR) under somehow stringent conditions. Restrictive conditions are placed to reduce the size of the search space, however, they also make the model inappropriate for the real-business world. In this paper, we consider a class of FLA problems that can assume more realistic conditions in real-life applications. A hybrid method of genetic algorithm and subgradient technique is used to solve the problem efficiently. [Copyright &y& Elsevier]

In this paper, based on the inherent characteristic of the contention relation between flows in ad hoc networks, we introduce the notion of the link’s interference set, extend the utility maximization problem representing congestion control in wireline networks to ad hoc networks, apply the penalty function approach and the subgradient method to solve this problem, and propose the congestion control algorithm Penalty function-based Optical Congestion Control (POCC) which is implemented in NS2 simulator. Specifically, each link transmits periodically the information on its congestion state to its interference set; the session at each source adjusts the transmission rate based on the optimal tradeoff between the utility value and the congestion level which the interference set of the links that this session goes though suffers from. MATLAB-based simulation results showed that POCC can approach the globally optimal solution. The NS2-based simulation results showed that POCC outperforms default TCP and ATCP to achieve efficient and fair resource allocation in ad hoc networks. [ABSTRACT FROM AUTHOR]

Abstract: A real function is called radially -differentiable at the point x if, for every real number h, the finite limit of the ratio exists whenever r tends to zero through the positive rationals. We establish that, in particular, Jensen-convex functions are everywhere radially -differentiable. Moreover, if f is Jensen-convex, then, for each x, the mapping is subadditive, and it is an upper bound for any additive mapping A satisfying the inequality for every y. We also characterize all set-valued mappings built up from additive solutions A of this inequality with some Jensen-convex function f. [Copyright &y& Elsevier]

A novel blind source separation approach and the corresponding adaptive algorithm is presented. It is assumed that the observation mixture is obtained through an unknown memoryless linear mapping of independent and bounded magnitude sources. We further assume an initial adaptive prewhitening of the original observation vector which transforms it into a white vector with the same dimension as the original source vector. Our approach is centered around the basic geometric fact that, under a certain boundedness assumption, the unitary mapping which transforms the whitening output vector into an independent vector has the minimum value of maximum (real component) magnitude output over the ensemble of all output components. Therefore, the related criterion is the minimization of the infinity norm of the real component of the unitary separator's output over all possible output combinations. For the minimization of the corresponding nondifferentiable cost function, we propose the use of subgradient optimization methods to obtain a low complexity iterative adaptive solution. The resulting algorithm is fairly intuitive and simple, and provides a low complexity solution especially to a class of multiuser digital communications problems. We provide examples at the end of this paper to illustrate the performance of our algorithm. [ABSTRACT FROM AUTHOR]