Faster, but weaker, relaxations for quadratically constrained quadratic programs

Abstract

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed blockdiagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.

Quadratic programming over ellipsoids with applications to constrained linear regression and tensor decomposition

“…Following this, we perform an alternating estimation process between x and α. We first initialize a unit-length vector x, compute α as in (8), then update x by solving (9), and update α again. The process is executed until there is no significant change in the object function value.…”

“…A more sophisticated problem is that of minimizing a convex quadratic function over an intersection of ellipsoids x T H m x = 1, bearing in mind that quadratic equalities characterize non-convex sets. The non-convex quadratic optimisation problem with quadratic equality constraints is known to be NP-hard [3,9,21]. Nevertheless, since gradients and Hessians of the constrained functions can be derived in an analytical form, the problem can be solved using interiorpoint algorithms for nonlinearly constrained minimization [17].…”

“…The QP problem with quadratic inequality constraints can also be solved efficiently using the Modified proportioning with gradient projections [12,13]. Some other common approaches are to convexify the problem using semidefinite relaxation techniques [15], second-order cone programming [18], or mixed SOCP-SDP relaxations [9]. For some particular cases, e.g., a quadratic function with two quadratic constraints, [6] shows that under a suitable assumption, the problem can be solved in polynomial time.…”

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

“…We now pass to an alternative put forward by Burer in his seminal paper [17], although this is not made explicit there in full generality; but see the more recent papers [19,20]. Basically, he concentrated on mixed-binary, linearly constrained quadratic optimization problems but extended the results to problems with additional quadratic equality constraints, e.g., complementarity constraints.…”

“…Due to their pivotal role for applications, bounds for this type of problem are currently receiving a lot of interest in the optimization community; for a nonexhaustive list, see [2,19,20,26,28,31,32,33,35,37,43,46].…”