We study exact semidefinite programming (SDP) relaxation for a given sparse quadratically constrained quadratic program (QCQP). The SDP relaxation is exact if, whenever it has an optimal solution, it admits a rank-at-most-one optimal solution that corresponds to an optimal solution of the QCQP. Using the maximal cliques of a chordal extension of the aggregate sparsity … Read more
We are interested in exactness, strong conic duality and dual attainability in copositive relaxations of quadratic optimization problems (QPs) of a special form, in which any (feasible) QP can be recast. By using our results, the celebrated Frank-Wolfe theorem on the attainability of any bounded QP even over unbounded polyhedra, regardless of whether the objective … Read more
Electricity storage is used for intertemporal price arbitrage and for ancillary services that balance unforeseen supply and demand fluctuations via frequency regulation. We present an optimization model that computes bids for both arbitrage and frequency regulation and ensures that storage operators can honor their market commitments at all times for all fluctuation signals in an … Read more
Recent LLM-driven discoveries have renewed interest in geometric packing problems. In this paper, we study several classes of such packing problems through the lens of modern global nonlinear optimization. Starting from comparatively direct nonlinear formulations, we consider packing circles in squares and fixed-perimeter rectangles, packing circles into minimum-area ellipses, packing regular polygons into regular polygons, … Read more
In this paper, we present a reformulation technique for convex mixed-integer nonlinear programming (MINLP) problems with nonseparable quadratic terms. For each convex non-diagonal matrix that defines quadratic expressions in the problem, we show that an eigenvalue or LDLT decomposition can be performed to transform the quadratic expressions into convex additively separable constraints. The reformulated problem … Read more
This paper studies exact semidefinite programming relaxations (SDPRs) for separable quadratically constrained quadratic programs (QCQPs). We consider the construction of a larger separable QCQP from multiple QCQPs with exact SDPRs. We show that exactness is preserved when such QCQPs are combined through a separable horizontal connection, where the coupling is induced through the right-hand-side parameters … Read more
We prove the NP-hardness, using Karp reductions, of some problems related to the correlation polytope and its corresponding cone, spanned by all of the n×n rank-one matrices over {0, 1}. The problems are: membership, rank of the decomposition, and a “relaxed rank” obtained from relaxing the zero-norm expression for the rank to an ℓ1 norm. … Read more
Motivated by the need for reliable traffic management under fluctuating travel demand, we study the problem of determining the worst-case congestion in a multi-commodity traffic network subject to demand uncertainty. To this end, we stress-test a given network by identifying demand realizations and corresponding travelers’ route choices that maximize congestion. The users of the traffic … Read more
We develop a successive, proximal difference-of-convex (DC) function penalty method for solving DC programs with DC constraints. The proposed approach relies on a DC penalty function that measures the violation of constraints and leads to a penalty reformulation sharing the same solution set as the original problem. The resulting penalty problem is a DC program … Read more
We present an effective warm-starting scheme for solving large linear complementarity problems (LCPs) arising from Nash equilibrium problems. The approach generates high-quality starting points that, when passed to the PATH solver, yield substantial reductions in computational time and variance. Our warm-start routine reformulates each agent’s LP using strong duality, leading to a master problem with … Read more