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.
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.
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.
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.
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.
We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed ...
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We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.
The nonconvex programming problem of minimizing a quasi-concave function over an efficient (or weakly efficient) set of a multiobjective linear program is studied. A cutting plane algorithm which finds an approximate ...
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The nonconvex programming problem of minimizing a quasi-concave function over an efficient (or weakly efficient) set of a multiobjective linear program is studied. A cutting plane algorithm which finds an approximate optimal solution in a finite number of steps is developed. For the particular ''all linear'' case the algorithm performs better, finding an optimal solution in a finite time, and being more easily implemented.
The Fuzzy clustering (FC) problem is a non-convex mathematical program which usually possesses several local minima. The global minimum solution of the problem is found using a simulated annealing-based algorithm. Som...
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The Fuzzy clustering (FC) problem is a non-convex mathematical program which usually possesses several local minima. The global minimum solution of the problem is found using a simulated annealing-based algorithm. Some preliminary computational experiments are reported and the solution is compared with that generated by the Fuzzy C-means algorithm.
Various classes of d.c. programs have been studied in the recent literature due to their importance in applicative problems. In this paper we consider a branch and bound approach for solving a class of d.c. problems. ...
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Various classes of d.c. programs have been studied in the recent literature due to their importance in applicative problems. In this paper we consider a branch and bound approach for solving a class of d.c. problems. Both stack policies and partitioning rules are analyzed, pointing out their performance effectiveness by means of the results of a computational experience. (C) 2010 Elsevier B.V. All rights reserved.
This letter studies joint transmit beamforming and antenna selection at a secondary base station (BS) with multiple primary users (PUs) in an underlay cognitive radio multiple-input single-output broadcast channel. Th...
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This letter studies joint transmit beamforming and antenna selection at a secondary base station (BS) with multiple primary users (PUs) in an underlay cognitive radio multiple-input single-output broadcast channel. The objective is to maximize the sum rate subject to the secondary BS transmit power, minimum required rates for secondary users, and PUs' interference power constraints. The utility function of interest is nonconcave and the involved constraints are nonconvex, so this problem is hard to solve. Nevertheless, we propose a new iterative algorithm that finds local optima at the least. We use an inner approximation method to construct and solve a simple convex quadratic program of moderate dimension at each iteration of the proposed algorithm. Simulation results indicate that the proposed algorithm converges quickly and outperforms existing approaches.
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