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We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its…
This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the…
A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh as a unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems.…
Recently, minimax optimization received renewed focus due to modern applications in machine learning, robust optimization, and reinforcement learning. The scale of these applications naturally leads to the use of first-order methods.…
This paper reports a novel result: with proper robot models on matrix Lie groups, one can formulate the kinodynamic motion planning problem for rigid body systems as \emph{exact} polynomial optimization problems that can be relaxed as…
The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to…
We show {\it semidefinite programming} (SDP) feasibility problem is equivalent to solving a {\it convex hull relaxation} (CHR) for a finite system of quadratic equations. On the one hand, this offers a simple description of SDP. On the…
This study considers the control problem with signal temporal logic (STL) specifications. Prior works have adopted smoothing techniques to address this problem within a feasible time frame and solve the problem by applying sequential…
In this article, we address a class of non convex, integer, non linear mathematical programs using dynamic programming. The mathematical program considered, whose properties are studied in this article, may be used to model the optimal…
Computational guidance is an emerging and accelerating trend in aerospace guidance and control. Combining machine learning and convex optimization, this paper presents a real-time computational guidance method for the 6-degrees-of-freedom…
We present an accelerated relax-and-round algorithm for concave coverage problems, which generalize the classic maximum coverage problem. Building on the relax-and-round framework of Barman et al. [STACS 2021], we propose two significant…
The tensor rank decomposition, also known as canonical polyadic(CP) or simply tensor decomposition, has a long history in multilinear algebra. However, computing a rank decomposition becomes particularly challenging when the rank lies…
Optimization problems with norm-bounding constraints arise in a variety of applications, including portfolio optimization, machine learning, and feature selection. A common approach to these problems involves relaxing the norm constraint…
In the first part of this work [32], we introduce a convex parabolic relaxation for quadratically-constrained quadratic programs, along with a sequential penalized parabolic relaxation algorithm to recover near-optimal feasible solutions.…
Adversarial training is well-known to produce high-quality neural network models that are empirically robust against adversarial perturbations. Nevertheless, once a model has been adversarially trained, one often desires a certification…
The Sparse Generalized Eigenvalue Problem (sGEP), a pervasive challenge in statistical learning methods including sparse principal component analysis, sparse Fisher's discriminant analysis, and sparse canonical correlation analysis,…
Semidefinite programs (SDPs) are a powerful framework for convex optimization and for constructing strong relaxations of hard combinatorial problems. However, solving large SDPs can be computationally expensive, motivating the use of…
We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. Mor\'e and G.…
This paper studies a structured compound stochastic program (SP) involving multiple expectations coupled by nonconvex and nonsmooth functions. We present a successive convex-programming based sampling algorithm and establish its…
Complex-variable matrix optimization problems (CMOPs) in Frobenius norm emerge in many areas of applied mathematics and engineering applications. In this letter, we focus on solving CMOPs by iterative methods. For unconstrained CMOPs, we…