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 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.

[1]  Tao Pham Dinh,et al.  DC Programming Approaches for BMI and QMI Feasibility Problems , 2014, ICCSAMA.

[2]  Joaquim Júdice,et al.  Solving the Quadratic Eigenvalue Complementarity Problem by DC Programming , 2015, MCO.

[3]  Le Thi Hoai An,et al.  DC Programming and DCA for General DC Programs , 2014, ICCSAMA.

[4]  Gert Wanka,et al.  Duality for multiobjective optimization problems with convex objective functions and D.C. constraints , 2006 .

[5]  J. Toland.  Duality in nonconvex optimization , 1978 .

[6]  Juan Enrique Martínez-Legaz,et al.  Duality in D.C. Programming: The Case of Several D.C. Constraints , 1999 .

[7]  Boris S. Mordukhovich,et al.  Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs , 2010, Math. Program..

[8]  Michel Volle,et al.  Concave duality: Application to problems dealing with difference of functions , 1988, Math. Program..

[9]  Masakazu Kojima,et al.  Sparsity in sums of squares of polynomials , 2005, Math. Program..

[10]  Vaithilingam Jeyakumar,et al.  Characterizing robust set containments and solutions of uncertain linear programs without qualifications , 2010, Oper. Res. Lett..

[11]  V. Jeyakumar,et al.  Characterizing Set Containments Involving Infinite Convex Constraints and Reverse-Convex Constraints , 2002, SIAM J. Optim..

[12]  Amir Ali Ahmadi,et al.  A Complete Characterization of the Gap between Convexity and SOS-Convexity , 2011, SIAM J. Optim..

[13]  Chong Li,et al.  Stable and Total Fenchel Duality for Convex Optimization Problems in Locally Convex Spaces , 2009, SIAM J. Optim..

[14]  B. Reznick.  Some concrete aspects of Hilbert's 17th Problem , 2000 .

[15]  Vaithilingam Jeyakumar,et al.  Robust SOS-convex polynomial optimization problems: exact SDP relaxations , 2013, Optimization Letters.

[16]  Nguyen Dinh,et al.  Dual Characterizations of Set Containments with Strict Convex Inequalities , 2006, J. Glob. Optim..

[17]  S. Kim,et al.  Semidefinite programming relaxation methods for global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets , 2016, J. Glob. Optim..

[18]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[19]  Diethard Klatte,et al.  A Frank–Wolfe Type Theorem for Convex Polynomial Programs , 2002, Comput. Optim. Appl..

[20]  Amir Ali Ahmadi,et al.  A convex polynomial that is not sos-convex , 2009, Mathematical Programming.

[21]  Vaithilingam Jeyakumar,et al.  New Sequential Lagrange Multiplier Conditions Characterizing Optimality without Constraint Qualification for Convex Programs , 2003, SIAM J. Optim..

[22]  Bernard Lemaire,et al.  Duality in Reverse Convex Optimization , 1998, SIAM J. Optim..

[24]  Masakazu Muramatsu,et al.  Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity , 2004 .

[25]  Le Thi Hoai An,et al.  Solving a Class of Linearly Constrained Indefinite Quadratic Problems by D.C. Algorithms , 1997, J. Glob. Optim..

[26]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[27]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[28]  Vaithilingam Jeyakumar,et al.  Generalized Farkas’ lemma and gap-free duality for minimax DC optimization with polynomials and robust quadratic optimization , 2016, J. Glob. Optim..

[29]  Bernard Lemaire,et al.  A General Duality Scheme for Nonconvex Minimization Problems with a Strict Inequality Constraint , 1998, J. Glob. Optim..

[30]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[31]  Le Thi Hoai An,et al.  The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems , 2005, Ann. Oper. Res..

[32]  Vaithilingam Jeyakumar,et al.  Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization , 2013, Mathematical Programming.

[33]  Vaithilingam Jeyakumar,et al.  Exact SDP relaxations for classes of nonlinear semidefinite programming problems , 2012, Oper. Res. Lett..

[34]  Lagrange-type duality in DC programming , 2014 .

[35]  Jiawang Nie,et al.  Polynomial Matrix Inequality and Semidefinite Representation , 2009, Math. Oper. Res..

[36]  Vaithilingam Jeyakumar,et al.  A new class of alternative theorems for SOS-convex inequalities and robust optimization , 2015 .

[38]  Jean B. Lasserre,et al.  Convexity in SemiAlgebraic Geometry and Polynomial Optimization , 2008, SIAM J. Optim..

[39]  Vaithilingam Jeyakumar,et al.  Sequential Lagrangian Conditions for Convex Programs with Applications to Semidefinite Programming , 2005 .

[40]  B. Reznick.  Extremal PSD forms with few terms , 1978 .

[41]  J. William Helton,et al.  Semidefinite representation of convex sets , 2007, Math. Program..

[42]  Radu Ioan Bot,et al.  Some new Farkas-type results for inequality systems with DC functions , 2007, J. Glob. Optim..

[43]  Vaithilingam Jeyakumar,et al.  Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs , 2013, Journal of Optimization Theory and Applications.

[44]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .