Electric power infrastructure faces increasing risk of damage and disruption due to wildfire. Operators of power grids in wildfire-prone regions must consider the potential impacts of unpredictable fires. However, traditional wildfire models do not effectively describe worst-case, or even high-impact, fire behavior. To address this issue, we propose a mixed-integer conic program to characterize an … Read more
We develop tractable convex relaxations for rank-constrained quadratic optimization problems over $n \times m$ matrices, a setting for which tractable relaxations are typically only available when the objective or constraints admit spectral (permutation-invariant) structure. We derive lifted semidefinite relaxations that do not require such spectral terms. Although a direct lifting introduces a large semidefinite constraint … Read more
A stochastic program typically involves several parameters, including deterministic first-stage parameters and stochastic second-stage elements that serve as input data. These programs are re-solved whenever any input parameter changes. However, in practical applications, quick decision-making is necessary, and solving a stochastic program from scratch for every change in input data can be computationally costly. This … Read more
For mixed-integer linear programming and linear programming it is well known that symmetries can have a negative impact on the performance of branch-and-bound and linear optimization algorithms. A common strategy to handle symmetries in linear programs is to reduce the dimension of the linear program by aggregating symmetric variables and solving a linear program of … Read more
Quasinormality is a classical constraint qualification originally introduced by Hestenes in 1975 and subsequently extensively studied in nonlinear programming and in problems with abstract constraints. In this paper, we extend this concept to the setting of nonlinear semidefinite programming (NSDP). We show that the proposed condition is strictly weaker than Robinson’s constraint qualification, while still … Read more
We focus on copositive and completely positive cones over symmetric cones of rank at least $5$, and in particular investigate whether these cones are spectrahedral shadows. We extend known results for nonnegative orthants of dimension at least $5$ to general symmetric cones of rank at least $5$. Specifically, we prove that when the rank of … Read more
Algorithms based on polyhedral outer approximations provide a powerful approach to solving mixed-integer nonlinear optimization problems. An initial relaxation of the feasible set is strengthened by iteratively adding linear inequalities and separating infeasible points. However, when the constraints are nonconvex, computing such separating hyperplanes becomes challenging. In this article, the moment-/sums-of-squares hierarchy is used in … Read more
We introduce a new class of semidefinite programming (SDP) relaxations for sparse box-constrained quadratic programs, obtained by a novel integration of the Reformulation Linearization Technique into standard SDP relaxations while explicitly exploiting the sparsity of the problem. The resulting relaxations are not implied by the existing LP and SDP relaxations for this class of optimization … Read more
Deploying mathematical optimization in autonomous production systems requires precise contracts for objects returned by an optimization solver. Unfortunately, conventions on dual solution and infeasibility certificates (rays) vary widely across solvers and classes of problems. This paper presents the theoretical framework used by MathOpt (a domain-specific language developed and used at Google) to unify these notions. … Read more
In modern Formula~1, strict regulations and highly optimized cars limit performance gains through hardware, increasing the importance of strategic decision-making. This work tackles the problem of computing a race strategy that minimizes total race time by jointly optimizing tire stints, compound selection, fuel load, and Energy Recovery System (ERS) deployment. We present a high-performance simulation … Read more