The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagr...
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The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semi- infinite optimization.
The paper investigates the long time average of the solutions of Hamilton-Jacobi equations with a noncoercive. nonconvex Hamiltonian in the torus R-2/Z(2). We give nonresonance conditions under which the long-time ave...
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The paper investigates the long time average of the solutions of Hamilton-Jacobi equations with a noncoercive. nonconvex Hamiltonian in the torus R-2/Z(2). We give nonresonance conditions under which the long-time average converges to a constant. In the resonant case, we show that the limit still exists, although it is nonconstant in general. We compute the limit at points where it is not locally constant. (C) 2009 Elsevier Masson SAS. All rights reserved.
Bilevel programming is characterized by two optimization problems located at different levels, in which the constraint region of the upper level problem is implicitly determined by the lower level problem. This paper ...
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Bilevel programming is characterized by two optimization problems located at different levels, in which the constraint region of the upper level problem is implicitly determined by the lower level problem. This paper is focused on a class of bilevel programming with a linear lower level problem and presents a new algorithm for solving this kind of problems by combining an evolutionary algorithm with the duality principle. First, by using the prime-dual conditions of the lower level problem, the original problem is transformed into a single-level nonlinear programming problem. In addition, for the dual problem of the lower level, the feasible bases are taken as individuals in population. For each individual, the values of dual variables can be obtained by taking the dual problem into account, thus simplifying the single-level problem. Finally, the simplified problem is solved, and the objective value is taken as the fitness of the individual. Besides, when nonconvex functions are involved in the upper level, a coevolutionary scheme is incorporated to obtain global optima. In the computational experiment, 10 problems, smaller or larger-scale, are solved, and the results show that the proposed algorithm is efficient and robust.
We present improvements to branch and bound techniques for globally optimizing functions with Lipschitz continuity properties by developing novel bounding procedures and parallelisation strategies. The bounding proced...
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We present improvements to branch and bound techniques for globally optimizing functions with Lipschitz continuity properties by developing novel bounding procedures and parallelisation strategies. The bounding procedures involve nonconvex quadratic or cubic lower bounds on the objective and use estimates of the spectrum of the Hessian or derivative tensor, respectively. As the nonconvex lower bounds are only tractable if solved over Euclidean balls, we implement them in the context of a recent branch and bound algorithm (Fowkes et al. in J Glob Optim 56:1791-1815, 2013) that uses overlapping balls. Compared to the rectangular tessellations of traditional branch and bound, overlapping ball coverings result in an increased number of subproblems that need to be solved and hence makes the need for their parallelization even more stringent and challenging. We develop parallel variants based on both data- and task-parallel paradigms, which we test on an HPC cluster on standard test problems with promising results.
In several methods of multiattribute decision making, pairwise comparison matrices are applied to derive implicit weights for a given set of decision alternatives. A class of the approaches is based on the approximati...
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In several methods of multiattribute decision making, pairwise comparison matrices are applied to derive implicit weights for a given set of decision alternatives. A class of the approaches is based on the approximation of the pairwise comparison matrix by a consistent matrix. In the paper this approximation problem is considered in the least-squares sense. In general, the problem is nonconvex and difficult to solve, since it may have several local optima. In the paper the classic logarithmic transformation is applied and the problem is transcribed into the form of a separable programming problem based on a univariate function with special properties. We give sufficient conditions of the convexity of the objective function over the feasible set. If such a sufficient condition holds, the global optimum of the original problem can be obtained by local search, as well. For the general case, we propose a branch-and-bound method. Computational experiments are also presented.
We describe a primal-dual application of the proximal point algorithm to nonconvex minimization problems. Motivated by the work of Spingarn and more recently by the work of Hamdi et al. about the primal resource-direc...
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We describe a primal-dual application of the proximal point algorithm to nonconvex minimization problems. Motivated by the work of Spingarn and more recently by the work of Hamdi et al. about the primal resource-directive decomposition scheme to solve nonlinear separable problems. This paper discusses some local results of a primal-dual regularization approach that leads to a decomposition algorithm. (C) 2004 Elsevier Inc. All rights reserved.
In Ref. 1, a general class of branch-and-bound methods was proposed by Horst for solving global optimization problems. One of the main contributions of Ref. 1 was the opportunity of handling partition elements whose f...
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In Ref. 1, a general class of branch-and-bound methods was proposed by Horst for solving global optimization problems. One of the main contributions of Ref. 1 was the opportunity of handling partition elements whose feasibility is not known. Deletion-by-infeasibility rules were presented for problems where the feasible set is convex, is defined by finitely many convex and reverse convex constraints, or is defined by Lipschitzian inequalities. In this note, we propose a new deletion-by-infeasibility rule for problems whose feasible set is defined by functions representable as differences of convex functions.
Deterministic nonconvex optimization solvers generate convex relaxations of noncon-vex functions by making use of underlying factorable representations. One approach introduces auxiliary variables assigned to each fac...
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Deterministic nonconvex optimization solvers generate convex relaxations of noncon-vex functions by making use of underlying factorable representations. One approach introduces auxiliary variables assigned to each factor that lifts the problem into a higher-dimensional decision space. In contrast, a generalized McCormick relaxation approach offers the significant advantage of constructing relaxations in the lower dimensionality space of the original problem without introducing auxiliary variables, often referred to as a "reduced-space" approach. Recent contributions illustrated how additional nontrivial inequality constraints may be used in factorable programming to tighten relaxations of the ubiquitous bilinear term. In this work, we exploit an analogous representation of McCormick relaxations and factorable programming to formulate tighter relaxations in the original decision space. We develop the under-lying theory to generate necessarily tighter reduced-space McCormick relaxations when a priori convex/concave relaxations are known for intermediate bilinear terms. We then show how these rules can be generalized within a McCormick relaxation scheme via three different approaches: the use of a McCormick relaxations coupled to affine arithmetic, the propagation of affine relaxations implied by subgradients, and an enumerative approach that directly uses relaxations of each factor. The developed approaches are benchmarked on a library of optimization problems using the *** optimizer. Two case studies are also considered to demonstrate the developments: an application in advanced manufacturing to optimize supply chain quality metrics and a global dynamic optimization application for rigorous model validation of a kinetic mechanism. The presented subgradient method leads to an improvement in CPU time required to solve the considered problems to is an element of-global optimality.
A general branch-and-bound conceptual scheme for global optimization is presented that includes along with previous branch-and-bound approaches also grid-search techniques. The corresponding convergence theory, as wel...
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A general branch-and-bound conceptual scheme for global optimization is presented that includes along with previous branch-and-bound approaches also grid-search techniques. The corresponding convergence theory, as well as the question of restart capability for branch-and-bound algorithms used in decomposition or outer approximation schemes are discussed. As an illustration of this conceptual scheme, a finite branch-and-bound algorithm for concave minimization is described and a convergent branch-and-bound algorithm, based on the previous one, is developed for the minimization of a difference of two convex functions.
We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-T...
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We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a quadratic programming problem. This method is a combination of the local algorithm DCA in DC programming with a branch and bound scheme well known in discrete and global optimization. Computational results on a class of quadratic bilevel programs are reported.
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