New Results on Quadratic Minimization

In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case in which the constraints are two quadratic inequalities. This formulation is termed the extended trust region subproblem in this paper, to distinguish it from the ordinary trust region subproblem, in which the constraint is a single ellipsoid. The computational complexity of the extended trust region subproblem in general is still unknown. In this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value, and consequently we obtain polynomial-time procedures for solving those special cases of quadratic optimization. For the extended trust region subproblem itself, we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution. Combining this with a result obtained in the first part of the paper, we propose a polynomial-time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint.

[1]  F. Hausdorff.  Der Wertvorrat einer Bilinearform , 1919 .

[2]  G. A. Miller,et al.  MATHEMATISCHE ZEITSCHRIFT. , 1920, Science.

[3]  Richard A. Tapia,et al.  A trust region strategy for nonlinear equality constrained op-timization , 1984 .

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

[5]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[6]  H. Wolkowicz,et al.  Trust Region Problems and Nonsymmetric Eigenvalue Perturbations , 1994 .

[7]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[8]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

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

[10]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[11]  Mihir Bellare,et al.  The complexity of approximating a nonlinear program , 1995, Math. Program..

[12]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[13]  Leonid Khachiyan,et al.  On the Complexity of Semidefinite Programs , 1997, J. Glob. Optim..

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

[15]  Y. Nesterov.  Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

[16]  Zhi-Quan Luo,et al.  Approximation Algorithms for Quadratic Programming , 1998, J. Comb. Optim..

[17]  Y. Nesterov.  Global quadratic optimization via conic relaxation , 1998 .

[18]  Tamás Terlaky,et al.  On maximization of quadratic form over intersection of ellipsoids with common center , 1999, Math. Program..

[19]  Yinyu Ye,et al.  Approximating quadratic programming with bound and quadratic constraints , 1999, Math. Program..

[20]  C. Roos,et al.  Theory and algorithms for semidefinite programming , 2000 .

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

[22]  XiaojiaoTong,et al.  A TRUST—REGION ALGORITHM FOR NONLINEAR INEQUALITY CONSTRAINED OPTIMIZATION , 2003 .