Abstract
In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.
Highlights
Consider the following generalized trust-region subproblem with additional linear inequality constraints: p∗ := inf q1 ( x ) := x T Ax + 2a T x q2 ( x ) := x T Bx + 2b T x + β ≤ 0, ciT x ≤ di, (1)i = 1, ..., m, where A, B ∈ Rn×n are symmetric matrices but not necessarily positive semidefinite, a, b, ci ∈ Rn and β, di ∈ R, i = 1, ..., m
We introduced two convex quadratic relaxations (CQRs) corresponding to two different conditions for model problem (1) that are the problems of minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints
We presented sufficient conditions based on an optimal solution of the CQRs under which the problem (1) is equivalent to exactly one of the CQRs
Summary
Consider the following generalized trust-region subproblem with additional linear inequality constraints: p∗ := inf q1 ( x ) := x T Ax + 2a T x q2 ( x ) := x T Bx + 2b T x + β ≤ 0, ciT x. The GTRS has been well studied in the literature and several methods have been proposed to solve it under various assumptions [11,18,19,20,21,22,23,24,25] It has strong duality and exact SDO-relaxation under the Slater condition [11,22]. They generalized this result to GTRS with two quadratic inequality constraints They showed that under certain additional conditions, the optimal solution of the original problem can be recovered from the optimal solution of the SOCP relaxation [26]. (ii) Exploiting the results in (i), we derive sufficient conditions that are expressed in terms of the data of the model problem (1) for exactness of the CQRs, strong Lagrangian duality and for tightness of the SDO-relaxation.
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