Showing papers in "Optimization in 2013"
TL;DR: In this article, a hybrid extragradient iteration method for finding a common element of the set of fixed points of a nonexpansive mapping and a set of solutions of equilibrium problems for a pseudomonotone and Lipschitz-type continuous bifunction is presented.
Abstract: In this article, we present a new hybrid extragradient iteration method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a pseudomonotone and Lipschitz-type continuous bifunction. We obtain strongly convergent theorems for the sequences generated by these processes in a real Hilbert space.
111 citations
TL;DR: In this paper, the authors obtained scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.
Abstract: Our main concern in this paper are concepts of nondominatedness w.r.t. a variable ordering structure introduced by P.L.Yu in 1974. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop-Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.
58 citations
TL;DR: It is shown that many different concepts of robustness and of stochastic programming can be described as special cases of a general non-linear scalarization method by choosing the involved parameters and sets appropriately, which leads to a unifying concept which can be used to handle robust and Stochastic optimization problems.
Abstract: We show that many different concepts of robustness and of stochastic programming can be described as special cases of a general non-linear scalarization method by choosing the involved parameters and sets appropriately. This leads to a unifying concept which can be used to handle robust and stochastic optimization problems. Furthermore, we introduce multiple objective (deterministic) counterparts for uncertain optimization problems and discuss their relations to well-known scalar robust optimization problems by using the non-linear scalarization concept. Finally, we mention some relations between robustness and coherent risk measures.
47 citations
TL;DR: In this paper, an alternating direction-based contraction-type method is developed to solve the general case of the linearly constrained separable convex programming problems whose involved operator is separable into finitely many individual functions.
Abstract: The classical alternating direction method (ADM) has been well studied in the context of linearly constrained convex programming and variational inequalities where the involved operator is formed as the sum of two individual functions without crossed variables. Recently, ADM has found many novel applications in diversified areas, such as image processing and statistics. However, it is still not clear whether ADM can be extended to the case where the operator is the sum of more than two individual functions. In this article, we extend the spirit of ADM to solve the general case of the linearly constrained separable convex programming problems whose involved operator is separable into finitely many individual functions. As a result, an alternating direction-based contraction-type method is developed. The realization of tackling this class of problems broadens the applicable scope of ADM substantially.
41 citations
TL;DR: In this article, a projection algorithm for solving an equilibrium problem where the bifunction is pseudomonotone with respect to its solution set is proposed, which is further combined with a cutting technique for minimizing the norm over the solution set.
Abstract: We propose a projection algorithm for solving an equilibrium problem (EP) where the bifunction is pseudomonotone with respect to its solution set. The algorithm is further combined with a cutting technique for minimizing the norm over the solution set of an EP whose bifunction is pseudomonotone with respect to the solution set.
39 citations
TL;DR: This article reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems, and main properties of the hyperbolic smoothing function.
Abstract: In this article, an approach for solving finite minimax problems is proposed. This approach is based on the use of hyperbolic smoothing functions. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in general algebraic modelling system. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem.
37 citations
TL;DR: In this paper, the Euler discretization to a class of linear optimal control problems is analyzed and convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size.
Abstract: We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L 1-norm and with order 1/2 w.r.t. the L 2-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.
36 citations
TL;DR: An algorithm is designed for solving the inverse 1-median problem on trees where it is possible to change the weights of vertices or to reduce the length of an edge to zero in case of symmetric bounds on the vertex weights.
Abstract: We investigate the inverse 1-median problem on trees where it is possible to change the weights of vertices or to reduce the length of an edge to zero. An algorithm is designed for solving this problem in case of symmetric bounds on the vertex weights.
32 citations
TL;DR: In this article, the optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points were proved and proved optimality for problems of calculus of variations of this type without using the classical theory.
Abstract: We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory.
30 citations
TL;DR: In this article, the problem of determining the minimal cardinality of double resolving sets for prism graphs has been studied and it is proved that the minimum cardinality is equal to four if is even and equal to three if is odd.
Abstract: In this paper, we consider the problem of determining the minimal cardinality of double resolving sets for prism graphs . It is proved that the minimal cardinality is equal to four if is even and equal to three if is odd.
27 citations
TL;DR: In this article, the authors compared the performance of GA, an evolutionary algorithm using particle swarm optimization mechanism with GA operators, and tabu search (TS) in terms of required CPU time and obtained objective values.
Abstract: Cumulative capacitated vehicle routing problem (CCVRP) is an extension of the well-known capacitated vehicle routing problem, where the objective is minimization of sum of the arrival times at nodes instead of minimizing the total tour cost. This type of routing problem arises when a priority is given to customer needs or dispatching vital goods supply after a natural disaster. This paper focuses on comparing the performances of neighbourhood and population-based approaches for the new problem CCVRP. Genetic algorithm (GA), an evolutionary algorithm using particle swarm optimization mechanism with GA operators, and tabu search (TS) are compared in terms of required CPU time and obtained objective values. In addition, a nearest neighbourhood-based initial solution technique is also proposed within the paper. To the best of authors’ knowledge, this paper constitutes a base for comparisons along with GA, and TS for further possible publications on the new problem CCVRP.
TL;DR: In this paper, an alternating direction method was proposed for solving convex semidefinite optimization problems, which only computes several metric projections at each iteration, and convergence analysis was presented and numerical experiments in solving matrix completion problems were reported.
Abstract: An alternating direction method is proposed for solving convex semidefinite optimization problems. This method only computes several metric projections at each iteration. Convergence analysis is presented and numerical experiments in solving matrix completion problems are reported.
TL;DR: In this paper, a new definition of monotone bifunctions is given, which is a slight generalization of the original definition given by Blum and Oettli, but which is better suited for relating monotonicity of a bifunction to a monotonous operator.
Abstract: A new definition of monotone bifunctions is given, which is a slight generalization of the original definition given by Blum and Oettli, but which is better suited for relating monotone bifunctions to monotone operators. In this new definition, the Fitzpatrick transform of a maximal monotone bifunction is introduced so as to correspond exactly to the Fitzpatrick function of a maximal monotone operator in case the bifunction is constructed starting from the operator. Whenever the monotone bifunction is lower semicontinuous and convex with respect to its second variable, the Fitzpatrick transform permits to obtain results on its maximal monotonicity.
TL;DR: In this paper, an optimization method for a national-level highway project planning based on a modified genetic algorithm is proposed, which adds to the existing methods by integrating various planning elements into a single system.
Abstract: This paper proposes an optimization method for a national-level highway project planning based on a modified genetic algorithm. The proposed method adds to the existing methods by integrating various planning elements into a single system. A simulation model is used in order to determine the best investment strategy with regard to net present value, time deviation from the initial plan and discrepancy between available resources and investment costs by taking into account economical, social, traffic and political factors. The outcome is a project schedule with an optimized cash flow. The proposed method was tested using the example of the National Highway Programme in Slovenia.
TL;DR: An algorithm is derived that may be regarded as a pivot as well as interior point one that produces a sequence of interior aswell as boundary points, including vertices, until reaching a pair of exact primal and dual optimal solutions.
Abstract: Simplex algorithms governed by some pivot rule and interior point algorithms are two diverging and competitive types of algorithms for solving linear programming problems. The former moves on the underlying polyhedron, from vertex to adjacent vertex, along edges until an optimal vertex is reached while the latter approaches an optimal point by moving across interior of the polyhedron. In this article, we derive an algorithm that may be regarded as a pivot as well as interior point one. It produces a sequence of interior as well as boundary points, including vertices, until reaching a pair of exact primal and dual optimal solutions. We report encouraging computational results with dense implementation of the algorithm on a set of Netlib test problems.
TL;DR: In this paper, the notion of approximate solution suggested by Kutateladze is dealt with, and, utilizing different scalarization approaches, some necessary and sufficient conditions for ϵ-(strong, weak, proper) efficiency are provided.
Abstract: In this article, approximate solutions of multi-objective optimization problems are analysed. The notion of approximate solution suggested by Kutateladze is dealt with, and, utilizing different scalarization approaches, some necessary and sufficient conditions for ϵ-(strong, weak, proper) efficiency are provided. Almost all of the provided results are established without any convexity assumption.
TL;DR: In this paper, a BFGS method for solving symmetric nonlinear equations was proposed, which possesses some favorable properties: (a) the generated sequence of iterates is norm descent; (b) the generation sequence of the quasi-Newton matrix is positive definite and (c) this method possesses the global convergence and superlinear convergence.
Abstract: In this article, we propose a BFGS method for solving symmetric nonlinear equations. The presented method possesses some favourable properties: (a) the generated sequence of iterates is norm descent; (b) the generated sequence of the quasi-Newton matrix is positive definite and (c) this method possesses the global convergence and superlinear convergence. Numerical results show that the presented method is interesting.
TL;DR: In this paper, a generalization of the ECP algorithm to cover a class of non-differentiable Mixed-Integer NonLinear Programming problems is studied, where the objective function is first assumed to be linear but also -pseudoconvex case is considered.
Abstract: In this article, a generalization of the ECP algorithm to cover a class of nondifferentiable Mixed-Integer NonLinear Programming problems is studied. In the generalization constraint functions are required to be -pseudoconvex instead of pseudoconvex functions. This enables the functions to be nonsmooth. The objective function is first assumed to be linear but also -pseudoconvex case is considered. Furthermore, the gradients used in the ECP algorithm are replaced by the subgradients of Clarke subdifferential. With some additional assumptions, the resulting algorithm shall be proven to converge to a global minimum.
TL;DR: In this article, the authors established necessary conditions and sufficient conditions of optimality in the form of Pontryagin principles for infinite-horizon discrete-time optimal control problems governed by a difference inequation.
Abstract: We establish necessary conditions and sufficient conditions of optimality in the form of Pontryagin principles for infinite-horizon discrete-time optimal control problems governed by a difference inequation.
TL;DR: Two new semidefinite programming (SDP) models for protein homology detection are presented by using novel transformations of the MIP problem to reduce the problem size significantly compared with the existing SDP models.
Abstract: Protein homology detection is a core problem in bioinformatics that helps annotate protein structural and functional features. It can be naturally formed as a mixed integer programming (MIP) problem with semi-supervised support vector machines (SVMs), which are accurate discriminative methods for classification. This article presents two new semidefinite programming (SDP) models for protein homology detection by using novel transformations of the MIP problem. Both models reduce the problem size significantly compared with the existing SDP models. Numerical experiments show that our first SDP model outperforms other methods in terms of misclassification errors for both synthetic data and the real data from the Protein Classification Benchmark Collection.
TL;DR: An error bound for the equilibrium problem interms of the generalized D-gap function is constructed which gives a significant modification to the error bound given by Konnov et al.
Abstract: The equilibrium problem (EP) can be formulated as an unconstrained minimization problem through the D-gap function. We present a descent type algorithm for solving EP based on the generalized D-gap function. We discuss the convergence properties of the proposed algorithm under suitable assumptions while supporting our approach with appropriate examples. We construct an error bound for the equilibrium problem interms of the generalized D-gap function which gives a significant modification to the error bound given by Konnov et al.
TL;DR: In this article, the second-order optimality conditions in set-valued optimization are given as an empty intersection of certain tangent cones in the objective space, and the duality arguments are used to give new multiplier rules.
Abstract: In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed.
TL;DR: It is shown that every finite infimizer and hence every solution of a polyhedral convex set optimization problem contains a pre-solution.
Abstract: We provide a counterexample to the remark in Lohne and Schrage [An algorithm to solve polyhedral convex set optimization problems, Optimization 62 (2013), pp. 131-141] that every solution of a polyhedral convex set optimization problem is a pre-solution. A correct statement is that every solution of a polyhedral convex set optimization problem obtained by the algorithm SetOpt is a pre-solution. We also show that every finite infimizer and hence every solution of a polyhedral convex set optimization problem contains a pre-solution.
TL;DR: A new method is presented to solve linear semi-infinite programming based on the fact that the nonnegative polynomial on could be turned into a positive semi-definite system, so it can use the non negative polynomials to approximate the semi-Infinite constraint.
Abstract: In this paper, we present a new method to solve linear semi-infinite programming. This method bases on the fact that the nonnegative polynomial on could be turned into a positive semi-definite system, so we can use the nonnegative polynomials to approximate the semi-infinite constraint. Furthermore, we set up an approximate programming for the primal linear semi-infinite programming, and obtain an error bound between two programming problems. Numerical results show that our method is efficient.
TL;DR: An adaptive constraint handling technique is studied on a set of test problems with two evolutionary algorithms and results indicate that the proposed adaptive technique produces results with better quality in terms of objective function values and constraint violations.
Abstract: In global optimization with evolutionary algorithms constraint handling presents major difficulties, especially in the case of equality constraints. Several techniques have been proposed to overcome this difficulty. In this work an adaptive constraint handling technique is studied on a set of test problems with two evolutionary algorithms. The results indicate that the proposed adaptive technique produces results with better quality in terms of objective function values and constraint violations. The comparison was assessed by performance profiles based on a new metric that considers information both on objective function value and constraints violation.
TL;DR: The proposed clustering approach in this article employs common weights derived from a data envelopment analysis-like model to cluster the data with input and output items and the performance of the proposed approach is tested on a real problem.
Abstract: The problem of clustering objects into groups is a branch in statistical multivariate analysis. This problem is usually solved using a statistical heuristic or an optimization. This article addresses the issue of investigation of the relative efficiency of operational units while taking into consideration the size of units in the sample. This article proposes a two-step procedure allowing managements, first, to cluster the operational units into different clusters, and second, to evaluate each unit in its cluster. The clustering problem is formulated as a multi-objective linear program and the set of units are partitioned into k groups based on their size. The proposed clustering approach in this article employs common weights derived from a data envelopment analysis-like model to cluster the data with input and output items. The performance of the proposed approach is tested on a real problem.
TL;DR: In this article, the authors considered a nonsmooth multiobjective programming problem with inequality and set constraints, and they derived the strong Kuhn-Tucker necessary optimality conditions.
Abstract: We consider a nonsmooth multiobjective programming problem with inequality and set constraints. By using the notion of convexificator, we extend the Abadie constraint qualification, and derive the strong Kuhn-Tucker necessary optimality conditions. Some other constraint qualifications have been generalized and their interrelations are investigated.
TL;DR: A hybridization of the Hestenes–Stiefel and Dai–Yuan conjugate gradient methods using a quadratic relaxation of a hybrid CG parameter proposed by Dai and Yuan is suggested.
Abstract: To take advantage of the attractive features of the Hestenes–Stiefel and Dai–Yuan conjugate gradient (CG) methods, we suggest a hybridization of these methods using a quadratic relaxation of a hybrid CG parameter proposed by Dai and Yuan. In the proposed method, the hybridization parameter is computed based on a conjugacy condition. Under proper conditions, we show that our method is globally convergent for uniformly convex functions. We give a numerical comparison of the implementations of our method and two efficient hybrid CG methods proposed by Dai and Yuan using a set of unconstrained optimization test problems from the CUTEr collection. Numerical results show the efficiency of the proposed hybrid CG method in the sense of the performance profile introduced by Dolan and More.
TL;DR: In this article, the radial directional derivative and the subdifferential of proper lower semicontinuous functions of convex functions were shown to be equivalent to controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles.
Abstract: We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate ϵ-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous fun...
TL;DR: In this article, the equivalence between vector variational-like inequalities involving limiting subdifferential and vector optimization problems is studied under a pseudo-invexity condition, and some properties of pseudo invex functions are obtained.
Abstract: Some properties of pseudoinvex functions, defined by means of limiting subdifferential, are obtained. Furthermore, the equivalence between vector variational-like inequalities involving limiting subdifferential and vector optimization problems are studied under pseudoinvexity condition.