We consider the problem of minimizing a sparse nonconvex quadratic function over the unit hypercube. By developing an extension of the Reformulation Linearization Technique (RLT) to continuous quadratic sets, we propose a novel second-order cone (SOC) representable relaxation for this problem. By exploiting the sparsity of the quadratic function, we establish a sufficient condition under … Read more
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The opioid crisis has remained a major public health challenge in the United States for many years. This study develops a data-driven decision support framework to guide policymakers in allocating county-level budgets across multiple expenditure categories in order to address the opioid crisis. We compile and curate a detailed dataset on fiscal policy and opioid-related … Read more
We present a Julia-based interface to the precompiled HALLaR and cuHALLaR binaries for large-scale semidefinite programs (SDPs). Both solvers are established as fast and numerically stable, and accept problem data in formats compatible with SDPA and a new enhanced data format taking advantage of Hybrid Sparse Low-Rank (HSLR) structure. The interface allows users to load … Read more
We investigate polyconvexity of the double well function $f(X) := |X-X_1|^2|X-X_2|^2$ for given matrices $X_1, X_2 \in \R^{n \times n}$. Such functions are fundamental in the modeling of phase transitions in materials, but their non-convex nature presents challenges for the analysis of variational problems. We prove that $f$ is polyconvex if and only if the … Read more
In this paper, we present a theoretical analysis of augmented Lagrangian (AL) methods applied to mathematical programs with complementarity constraints (MPCCs). Our focus is on a variant that reformulates the complementarity constraints using slack variables, where these constraints are handled directly in the subproblems rather than being penalized. We introduce specialized constraint qualifications (CQs) of … Read more
We propose a first order algorithm, a modified version of FISTA, to solve an optimization problem with an objective function that is the sum of a possibly nonconvex function, with Lipschitz continuous gradient, and a convex function which can be nonsmooth. The algorithm is shown to have an iteration complexity of \(\mathcal{O}(\epsilon^{-2})\) to find an … Read more
Contextual optimization enhances decision quality by leveraging side information to improve predictions of uncertain parameters. However, existing approaches face significant challenges when dealing with multimodal or mixtures of distributions. The inherent complexity of such structures often precludes an explicit functional relationship between the contextual information and the uncertain parameters, limiting the direct applicability of parametric … Read more
We study a generalization of the classical discrete-time, Linear-Quadratic-Gaussian (LQG) control problem where the noise distributions affecting the states and observations are unknown and chosen adversarially from divergence-based ambiguity sets centered around a known nominal distribution. For a finite horizon model with Gaussian nominal noise and a structural assumption on the divergence that is satisfied … Read more
This paper presents a restart version of the Adaptive Moment Estimation (Adam) method for bound constrained nonlinear optimization problems. It aims to avoid getting trapped in a local minima and enable exploration the global optimum. The proposed method combines an adapted restart strategy coupling with barrier methodology to handle the bound constraints. Computational comparison with … Read more