On minimizing difference of a SOS-convex polynomial and a support function over a SOS-concave matrix polynomial constraint

In this paper, we establish tractable sum of squares characterizations of the containment of a convex set, defined by a SOS-concave matrix inequality, in a non-convex set, defined by difference of a SOS-convex polynomial and a support function, with Slater’s condition. Using our set containment char...

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Bibliographic Details
Published in:Mathematical programming Vol. 169; no. 1; pp. 177 - 198
Main Authors: Lee, Jae Hyoung, Lee, Gue Myung
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2018
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Containment
Convex analysis
Full Length Paper
Inequality
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Polynomials
Semidefinite programming
Theoretical
Strong duality
SOS-concave matrix
90C30
26A51
SOS-convex polynomials
90C22
Sums of squares polynomials
DC programming
Set containment
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:In this paper, we establish tractable sum of squares characterizations of the containment of a convex set, defined by a SOS-concave matrix inequality, in a non-convex set, defined by difference of a SOS-convex polynomial and a support function, with Slater’s condition. Using our set containment characterization, we derive a zero duality gap result for a DC optimization problem with a SOS-convex polynomial and a support function, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programs. Also, we present the relations of their solutions. Finally, through a simple numerical example, we illustrate our results. Particularly, in this example we find the optimal solution of the original problem by calculating the optimal solution of its associated semidefinite problem.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-017-1210-z