A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for a...
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A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n a parts per thousand yen 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic.
We propose a branch-and-bound algorithm for solving nonconvexquadratically-constrained quadratic programs. The algorithm is novel in that branching is done by partitioning the feasible region into the Cartesian produ...
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We propose a branch-and-bound algorithm for solving nonconvexquadratically-constrained quadratic programs. The algorithm is novel in that branching is done by partitioning the feasible region into the Cartesian product of two-dimensional triangles and rectangles. Explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second-order cone representable. The usefulness of these new relaxations is demonstrated both theoretically and computationally.
In this paper, we investigate a class of nonconvex quadratic programming with box constrains. A new branch and bound algorithm is proposed. The improvement of the new method is how to determine the lower bound. We put...
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ISBN:
(纸本)9781467352536
In this paper, we investigate a class of nonconvex quadratic programming with box constrains. A new branch and bound algorithm is proposed. The improvement of the new method is how to determine the lower bound. We put nonconvex quadratic programming into convex quadraticprogramming, and get an optimal solution as lower bound of original problem. Meanwhile, an upper bound is got by existing methods. Moreover, by used of the branch and bound algorithm, we can solve the original problem by solved a series of subproblems. Finally, the convergence of the proposed new algorithm is proved.
In/this paper, we discuss computational aspects of an interior- point algorithm [1] for indefinite quadraticprogramming problems with box constraints. The algorithm finds a local minimizer by successively solving ind...
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Battery energy storage systems (BESSs) are gaining attention due to reduced costs and high flexibility, but developing accurate models for operation presents challenges. This paper introduces a model for the charging ...
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ISBN:
(纸本)9798350315097
Battery energy storage systems (BESSs) are gaining attention due to reduced costs and high flexibility, but developing accurate models for operation presents challenges. This paper introduces a model for the charging and discharging processes via a single current decision variable, approximates the relation between the open circuit voltage and the state of charge with linear functions, and presents an optimization model with bilinear constraints for identifying optimal BESS operational strategies. A transformation technique is introduced to manage the bilinear constraints, transforming the model into an exponential optimization problem with linear constraints. A new sequential linear and quadraticprogramming approach is developed, with proven convergence. Preliminary experiments demonstrate the efficacy and efficiency of this approach.
A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for a...
详细信息
A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n a parts per thousand yen 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic.
In this paper we address the Phase Retrieval problem, which aims to recover the phase of the Fourier transform of a signal when only magnitude measurements are available. Following recent developments in Phase Retriev...
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ISBN:
(纸本)9781479970612
In this paper we address the Phase Retrieval problem, which aims to recover the phase of the Fourier transform of a signal when only magnitude measurements are available. Following recent developments in Phase Retrieval, the problem can be transformed into a convex semidefinite programming optimization problem, which can be solved using Matrix Completion techniques. In this paper the acquisition process is modeled using a likelihood function, which splits the original problem into two convex optimization problems, and alternates between the solution of each of them. To relate both convex problems we introduce a heuristic, which results in fast convergence of the proposed method.
The aim of this paper is to investigate the continuity and the directional differentiability of the value function in quadratically constrained nonconvex quadratic programming problem. Our result can be used in some c...
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The aim of this paper is to investigate the continuity and the directional differentiability of the value function in quadratically constrained nonconvex quadratic programming problem. Our result can be used in some cases where the existing results on differential stability in nonlinear programming (applied to quadraticprogramming) cannot be used.
A method is presented for the generation of test problems for global optimization algorithms. Given a bounded polyhedron in R and a vertex, the method constructs nonconvexquadratic functions (concave or indefinite) w...
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A method is presented for the generation of test problems for global optimization algorithms. Given a bounded polyhedron in R and a vertex, the method constructs nonconvexquadratic functions (concave or indefinite) whose global minimum is attained at the selected vertex. The construction requires only the use of linear programming and linear systems of equations.
An approximate combinatorial method for solving optimization problems is used to search for a global maximum of a quadratic function on a parallelepiped. Approximating functions in this method are majorants of an obje...
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An approximate combinatorial method for solving optimization problems is used to search for a global maximum of a quadratic function on a parallelepiped. Approximating functions in this method are majorants of an object function which are defined on subsets of a parallelepiped of admissible solutions. The method is based on a diagonal or block-diagonal LDLT-factorization of a matrix of an object function.
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