Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.

[1]  Yonina C. Eldar,et al.  Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints , 2006, SIAM J. Optim..

[2]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[3]  J. J. Moré.  Generalizations of the trust region problem , 1993 .

[4]  Boris Polyak.  Convexity of Quadratic Transformations and Its Use in Control and Optimization , 1998 .

[5]  Shuzhong Zhang,et al.  On Cones of Nonnegative Quadratic Functions , 2003, Math. Oper. Res..

[6]  Henry Wolkowicz,et al.  Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations , 1995, SIAM J. Optim..

[7]  Ya-Xiang Yuan,et al.  Optimality Conditions for the Minimization of a Quadratic with Two Quadratic Constraints , 1997, SIAM J. Optim..

[8]  Vaithilingam Jeyakumar,et al.  Alternative Theorems for Quadratic Inequality Systems and Global Quadratic Optimization , 2009, SIAM J. Optim..

[9]  Amir Beck,et al.  Convexity Properties Associated with Nonconvex Quadratic Matrix Functions and Applications to Quadratic Programming , 2009, J. Optimization Theory and Applications.

[10]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[11]  Henry Wolkowicz,et al.  Zero duality gaps in infinite-dimensional programming , 1990 .

[12]  Kees Roos,et al.  Robust Solutions of Uncertain Quadratic and Conic-Quadratic Problems , 2002, SIAM J. Optim..

[13]  Jean-Philippe Vial,et al.  Robust Optimization , 2021, ICORES.

[14]  Constantine Caramanis,et al.  Theory and Applications of Robust Optimization , 2010, SIAM Rev..

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

[16]  Vaithilingam Jeyakumar,et al.  A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty , 2011, Oper. Res. Lett..

[17]  Melvyn Sim,et al.  Robust linear optimization under general norms , 2004, Oper. Res. Lett..

[18]  Ya-Xiang Yuan,et al.  A trust region algorithm for equality constrained optimization , 1990, Math. Program..

[19]  L. L. Dines.  On the mapping of quadratic forms , 1941 .

[20]  N. Q. Huy,et al.  Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization , 2009 .

[21]  Ya-Xiang Yuan,et al.  On a subproblem of trust region algorithms for constrained optimization , 1990, Math. Program..

[22]  Tamás Terlaky,et al.  A Survey of the S-Lemma , 2007, SIAM Rev..

[23]  Samuel Burer,et al.  Second-Order-Cone Constraints for Extended Trust-Region Subproblems , 2013, SIAM J. Optim..

[24]  Vaithilingam Jeyakumar,et al.  Strong Duality in Robust Convex Programming: Complete Characterizations , 2010, SIAM J. Optim..

[25]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[26]  Laurent El Ghaoui,et al.  Robust Solutions to Least-Squares Problems with Uncertain Data , 1997, SIAM J. Matrix Anal. Appl..

[27]  Shuzhong Zhang,et al.  New Results on Quadratic Minimization , 2003, SIAM J. Optim..

[28]  Zhi-You Wu,et al.  Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions , 2007, Math. Program..