Convex Analysis: (pms-28)

This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization--a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We focus on examples having just a few variables or a few constraints for which the quadratic problem can be formulated as a copositive-style problem, which itself can be recast in terms of linear, second-order-cone, and semidefinite optimization. A particular highlight is the role played by the geometry of the feasible set.

/pdf/a-gentle-geometric-introduction-to-copositive-optimization-4plune9s6n.pdf

A gentle, geometric introduction to copositive optimization

We consider an optimization problem of the form[EQUATION]where P ⊆ Rn is a polyhedron defined by m inequalities and Q is general and the μh e Rn and the rh quantities are given. In the case |S| = 1, |K| = 0 and m = 0 one obtains the classical trust-region subproblem; a strongly NP-hard problem which has been the focus of much interest because of applications to combinatorial optimization and nonlinear programming. We prove that for each fixed pair |S| and |K| our problem can be solved in polynomial time provided that either (1) |K| > 0 and the number of faces of P that intersect [EQUATION] is polynomially bounded, or (2) |K| = 0 and m is bounded.

http://www.optimization-online.org/DB_FILE///2013/07/3951.pdf

Polynomial solvability of variants of the trust-region subproblem

Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. In this paper we review the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. Then we present applications of the theory to trust-region problems and signal processing.

/pdf/low-rank-semidefinite-programming-theory-and-applications-3ehedxp1w7.pdf

Low-Rank Semidefinite Programming: Theory and Applications

Recent research has focused on the complexity of extensions of the classical trust-region subproblem, which addresses the minimization of a quadratic function over a unit ball in ${\mathbb{R}}^n,$ ...

http://www.optimization-online.org/DB_FILE/2013/12/4170.pdf

A Note on Polynomial Solvability of the CDT Problem

This paper studies an extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with $$m$$ m linear inequality constraints. When $$m=0,\,m = 1$$ m = 0 , m = 1 , or $$m = 2$$ m = 2 and the linear constraints are parallel, it is known that the eTRS optimal value equals the optimal value of a particular convex relaxation, which is solvable in polynomial time. However, it is also known that, when $$m \ge 2$$ m ? 2 and at least two of the linear constraints intersect within the ball, i.e., some feasible point of the eTRS satisfies both linear constraints at equality, then the same convex relaxation may admit a gap with eTRS. This paper shows that the convex relaxation has no gap for arbitrary $$m$$ m as long as the linear constraints are non-intersecting.

http://www.optimization-online.org/DB_FILE/2013/02/3789.pdf

The trust region subproblem with non-intersecting linear constraints

The trust-region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite-programming (SDP) relaxation. Intersecting the feasible set with a second ellipsoid results in the two-trust-region subproblem (TTRS). Even though TTRS can also be solved in polynomial time, existing algorithms do not use SDP. In this paper, we investigate the use of SDP for TTRS. Starting from the basic SDP relaxation of TTRS, which admits a gap, recent research has tightened the basic relaxation using valid second-order-cone inequalities. Even still, closing the gap requires more. For the special case of TTRS in dimension $n=2$, we fully characterize the remaining valid inequalities, which can be viewed as strengthened versions of the second-order-cone inequalities just mentioned. We also demonstrate that these valid inequalities can be used computationally even when $n > 2$ to solve TTRS instances that were previously unsolved using SDP-based techniques.

http://www.optimization-online.org/DB_FILE/2013/11/4126.pdf

A Two-Variable Approach to the Two-Trust-Region Subproblem

Let $$\mathcal {F}$$
 be a quadratically constrained, possibly nonconvex, bounded set, and let $$\mathcal {E}_1, \ldots , \mathcal {E}_l$$
 denote ellipsoids contained in $$\mathcal {F}$$
 with non-intersecting interiors. We prove that minimizing an arbitrary quadratic $$q(\cdot )$$
 over $$\mathcal {G}:= \mathcal {F}{\setminus } \cup _{k=1}^\ell {{\mathrm{int}}}(\mathcal {E}_k)$$
 is no more difficult than minimizing $$q(\cdot )$$
 over $$\mathcal {F}$$
 in the following sense: if a given semidefinite-programming (SDP) relaxation for $$\min \{ q(x) : x \in \mathcal {F}\}$$
 is tight, then the addition of l linear constraints derived from $$\mathcal {E}_1, \ldots , \mathcal {E}_l$$
 yields a tight SDP relaxation for $$\min \{ q(x) : x \in \mathcal {G}\}$$
 . We also prove that the convex hull of $$\{ (x,xx^T) : x \in \mathcal {G}\}$$
 equals the intersection of the convex hull of $$\{ (x,xx^T) : x \in \mathcal {F}\}$$
 with the same l linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixed-integer nonconvex quadratic programming.

Boshi Yang

Papers

The trust region subproblem with non-intersecting linear constraints

A Two-Variable Approach to the Two-Trust-Region Subproblem

Quadratic programs with hollows

Asymptotically tight conic approximations for chance-constrained AC optimal power flow

Convex hull results on quadratic programs with non-intersecting constraints