The integer quadratic moment hull is the closed convex hull of the pairs \((z,zz^{\top})\) with \(z\in\mathbb{Z}^n\). We prove that one round of Chvátal–Gomory (CG) cuts applied to its basic semidefinite relaxation recovers this hull exactly if and only if \(n\le6\). Our approach identifies its facets with perfect Delaunay polyhedra and characterizes their recovery through depth, … Read more
This paper proposes a simple facet enumeration algorithm for convex polytopes. The complexity of the algorithm is analyzed. The algorithm is implemented in Matlab. Some polytopes with known H-representations and V-representations are used as test examples. The preliminary numerical test shows the effectiveness and efficiency of the proposed algorithm. Due to the duality between the … Read more
Linear inequalities can generate large constraint systems in semidefinite relaxations of binary quadratic programs, making the quadratic subproblems of splitting methods expensive to solve. We develop a box-cone representation of bound-factor reformulation-linearization technique (RLT) constraints in semidefinite relaxations of binary quadratic programs. These constraints arise by multiplying each original linear inequality by every binary variable … Read more
In a predictor-corrector arc-search method, a point on the predictor arc is first selected and a corrector is then computed at that point. Since the Karush-Kuhn-Tucker (KKT) matrix changes with the selected point, this correction may require a second factorization in each iteration. We propose a one-factorization arc-search method (OFAS) for semidefinite programming (SDP) that … Read more
We show that unless \(\mathrm P=\mathrm{NP}\), there cannot be a polynomial-time (or even pseudo-polynomial-time) algorithm for output feedback stabilization of a linear dynamical system with a linear controller. This settles one of the best-known open problems in control theory. The result holds in both continuous and discrete time. We also present a family of stabilizable … Read more
We consider solving (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) — with heuristic enhancements and GPU implementation — has been very successful in solving huge-scale linear optimization problems (LPs). However, its application to more general conic convex optimization problems is … Read more
We consider the Lorentz positive cone of \(n \times m\) matrices that map the Lorentz cone in \(R^m\) into the Lorentz cone in \(R^n\). The Lorentz positive cone and its dual, the cone of Lorentz separable matrices, are shown to provide polynomial-time algorithms for the problem of minimizing a bilinear objective over variables contained in … Read more
We present a Julia computational toolbox for linear optimization problems with joint affine chance constraints under elliptically symmetric uncertainty. The toolbox combines a spherical–radial oracle for estimating the joint probability and its gradient with three structured optimization methods: Proximal, Feasible, and Penalty. The oracle supports several elliptically symmetric distributions and is integrated with these methods … Read more
State-of-the-art exact methods for the Maximum Cut problem are based on solving linear and semidefinite programming relaxations embedded into a branch-and-bound algorithm. For linear programming formulations, it was shown recently that it is thereby sufficient to enforce the integrality of the variables associated with the edges of a spanning tree. Our first contribution is to … Read more
When linearizing binary quadratic programs, the most usual way is to replace bilinear products with additional variables constrained to take on consistent values in any feasible solution. In this setting, column generation is a principally desirable solution technique, for instance because the number of such additional linearization variables may be large while many of them … Read more