![]() ![]() Our analysis for objective value exactness and convex hull exactness stems from a geometric treatment of the projected SDP relaxation and crucially considers how the objective function interacts with the constraints. In particular, we will define and examine three notions of SDP exactness: (i) objective value exactness - the condition that the optimal value of the QCQP and the optimal value of its SDP relaxation coincide, (ii) convex hull exactness - the condition that the convex hull of the QCQP epigraph coincides with the (projected) SDP epigraph, and (iii) the rank-one generated (ROG) property - the condition that a particular conic subset of the positive semidefinite matrices related to a given QCQP is generated by its rank-one matrices. In this tutorial, we will study the SDP relaxation for general QCQPs, present various exactness concepts related to this relaxation and discuss conditions guaranteeing such SDP exactness. Although QCQPs are NP-hard to solve in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. Such problems arise naturally in many areas of operations research, computer science, and engineering. In a QCQP, we are asked to minimize a (possibly nonconvex) quadratic function subject to a number of (possibly nonconvex) quadratic constraints. Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems. ![]() We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared to when they are not. In our computational experiments, we separate many rounds of these inequalities starting from McCormick's relaxation of instances where each constraint is a separable bilinear constraint set. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. We first prove that the semi-definite programming relaxation provides no benefit over the McCormick relaxation for such problems. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. Recently, we proposed a class of inequalities called lifted bilinear cover inequalities, which are second-order cone representable convex inequalities, and are valid for a set described by a separable bilinear constraint together with bounds on variables. Our approach unifies andĮxtends previous results, and we illustrate its applicability and generality Precisely the corresponding conic/convex hulls. Under further assumptions, we prove that these two sets capture ![]() Under several easy-to-verify assumptions, we derive simple,Ĭomputable convex relaxations $K \cap S$ and $K \cap S \cap H$, where $S$ is a Nonconvex cone defined by a single homogeneous quadratic, and $H$ is an affine The form $K \cap Q$ and $K \cap Q \cap H$, where $K$ is a SOCr cone, $Q$ is a In this paper, we study more general intersections of U$, equals the intersection of $E$ with an additional second-order-cone For example, it has been shown-by several authors usingĭifferent techniques-that the convex hull of the intersection of anĮllipsoid, $E$, and a split disjunction, $(l - x_j)(x_j - u) \le 0$ with $l < Of course, we can also use function null to find an orthonormal basis for the null space of $\rm A$.A recent series of papers has examined the extension ofĭisjunctive-programming techniques to mixed-integer second-order-cone In MATLAB, we can use function rref to find permutation matrix $\rm P$ and matrix $\rm F$. Suppose we would like to find the intersection of $2$ hyperplanes in $\mathbb R^n$ ![]()
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