This paper is concerned with optimal power flow (OPF), which is the problem of optimizing the transmission of electricity in power systems. Our main contributions are as follows: (i) we propose a novel parabolic relaxation, which transforms non-convex OPF problems into convex quadratically-constrained quadratic programs (QCQPs) and can serve as an alternative to the common practice semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, (ii) we propose a penalization technique which is compatible with the SDP, SOCP, and parabolic relaxations and guarantees the recovery of feasible solutions for OPF, under certain assumptions. The proposed penalized convex relaxation can be used sequentially to find feasible and near-globally optimal solutions for challenging instances of OPF. Extensive numerical experiments on small and large-scale benchmark systems corroborate the efficacy of the proposed approach. By solving a few rounds of penalized convex relaxation, fully feasible solutions are obtained for benchmark test cases from [1]-[3] with as many as 13659 buses. In all cases, the solutions obtained are not more than 0.32% worse than the best-known solutions.
This paper is concerned with the unconstrained binary polynomial program (UBPP), which has a host of applications in many science and engineering fields. By leveraging the global exact penalty for its DC constrained SDP reformulation, we achieve an equivalent DC penalized SDP, and propose a continuous relaxation approach by seeking the critical point of the Burer-Monteiro factorization for a finite number of DC penalized SDPs with increasing penalty factors. A globally convergent majorization-minimization (MM) method with extrapolation is also developed to capture such critical points. Under a…
University of Texas Arlington（德克萨斯大学阿灵顿分校）；Cognitiv；University of California Berkeley（加州大学伯克利分校）
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. In this second part, we show that starting from a feasible solution or a near-feasible solution satisfying certain regularity conditions, the sequential penalized parabolic relaxation algorithm convergences to a point which satisfies Karush-Kuhn-Tucker optimality conditions. Next, we present numerical experiments on benchmark non-convex QCQP problems as well …
The first part of this paper proposed a family of penalized convex relaxations for solving optimization problems with bilinear matrix inequality (BMI) constraints. In this part, we generalize our approach to a sequential scheme which starts from an arbitrary initial point (feasible or infeasible) and solves a sequence of penalized convex relaxations in order to find feasible and near-optimal solutions for BMI optimization problems. We evaluate the performance of the proposed method on the H2 and Hinfinity optimal controller design problems with both centralized and decentralized structures. Th…
Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromi…
This two-part paper is concerned with the problem of minimizing a linear objective function subject to a bilinear matrix inequality (BMI) constraint. In this part, we first consider a family of convex relaxations which transform BMI optimization problems into polynomial-time solvable surrogates. As an alternative to the state-of-the-art semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, a computationally efficient parabolic relaxation is developed, which relies on convex quadratic constraints only. Next, we developed a family of penalty functions, which can be…
In this paper, we give a new penalized semidefinite programming approach for non-convex quadratically-constrained quadratic programs (QCQPs). We incorporate penalty terms into the objective of convex relaxations in order to retrieve feasible and near-optimal solutions for non-convex QCQPs. We introduce a generalized linear independence constraint qualification (GLICQ) criterion and prove that any GLICQ regular point that is sufficiently close to the feasible set can be used to construct an appropriate penalty term and recover a feasible solution. Inspired by these results, we develop a heurist…
This paper is concerned with the computation of the high-dimensional zero-norm penalized quantile regression estimator, defined as a global minimizer of the zero-norm penalized check loss function. To seek a desirable approximation to the estimator, we reformulate this NP-hard problem as an equivalent augmented Lipschitz optimization problem, and exploit its coupled structure to propose a multi-stage convex relaxation approach (MSCRA\_PPA), each step of which solves inexactly a weighted $\ell_1$-regularized check loss minimization problem with a proximal dual semismooth Newton method. Under a …
University of Texas, Arlington（德克萨斯大学阿灵顿分校）；Cognitiv；University of California, Berkeley（加州大学伯克利分校）
For general quadratically-constrained quadratic programming (QCQP), we propose a parabolic relaxation described with convex quadratic constraints. An interesting property of the parabolic relaxation is that the original non-convex feasible set is contained on the boundary of the parabolic relaxation. Under certain assumptions, this property enables one to recover near-optimal feasible points via objective penalization. Moreover, through an appropriate change of coordinates that requires a one-time computation of an optimal basis, the easier-to-solve parabolic relaxation can be made as strong a…
Concentration of measure-valued solutions for semilinear parabolic equations
Fédération ENAC ISAE-SUPAERO ONERA, Université de Toulouse（法国图卢兹大学）；LAAS-CNRS, Université de Toulouse（法国图卢兹大学）；Czech Technical University in Prague（布拉格捷克理工大学）
A Generic Branch-and-Bound Algorithm for $\ell_0$-Penalized Problems with Supplementary Material
Cl\' e ment Elvira CentraleSupelec, CNRS, IETR - UMR 6164, Rennes, France Th\' e o Guyard Scale-AI Chair, Polytechnique Montr\' e al, Canada C\' e dric Herzet Ensai, CNRS, CREST - UMR 9194, Rennes, France
This paper deals with the non-convex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows, for which existing methods lack guaranteed convergence and theoretical analysis. Motivating by these limitations, we propose a novel convexification framework for the PSSE using semidefinite programming (SDP) and second-order cone prog…
We consider the task of classification in the high dimensional setting where the number of features of the given data is significantly greater than the number of observations. To accomplish this task, we propose a heuristic, called sparse zero-variance discriminant analysis (SZVD), for simultaneously performing linear discriminant analysis and feature selection on high dimensional data. This method combines classical zero-variance discriminant analysis, where discriminant vectors are identified in the null space of the sample within-class covariance matrix, with penalization applied to induce …
In this article, we consider parabolic equations on a bounded open connected subset $Ω$ of $\R^n$. We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer? We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random …
In this work, a class of parabolic economic optimal control problems is considered. These problems are characterized by pointwise state constraints regularized by a parameter, which transforms the pure state constraints in mixed control-state ones. However, the convergence of classical (semismooth) Newton methods deteriorates for decreasing values of the regularization parameter. To tackle this problem, a nonlinear preconditioner is introduced. This is based on an overlapping optimized waveform-relaxation method characterized by Robin transmission conditions. Numerical experiments show that ap…
Computation of Least Trimmed Squares: A Branch-and-Bound framework with Hyperplane Arrangement Enhancements
MIT Operations Research Center（麻省理工学院运营研究中心）；USC Daniel J. Epstein Department of Industrial and Systems Engineering（南加州大学Daniel J. Epstein工业与系统工程系）；MIT Sloan School of Management, Operations Research Center（麻省理工学院斯隆管理学院，运营研究中心）
Regularization vs. Relaxation: A conic optimization perspective of statistical variable selection
Department of Mathematics, Washington State University（数学系，华盛顿州立大学）；Department of Statistics, University of Connecticut（统计系，康涅狄格大学）；Department of Industrial and Systems Engineering, University of Wisconsin-Madison（工业与系统工程系，威斯康星大学麦迪逊分校）
A penalty-type method for relaxed inverse optimal control problems
University of Siegen（锡根大学）；Higher Training College, University of Yaoundé I（雅温得第一大学高等培训学院）；AIMS-Cameroon Research Center（AIMS喀麦隆研究中心）；University of Marburg（马尔堡大学）
Scalable Bi-causal Optimal Transport via KL Relaxation and Policy Gradients
Department of Applied Mathematics and Statistics, Data Science and AI Institute, and Mathematical Institute for Data Science, Johns Hopkins University（应用数学与统计学系、数据科学与人工智能学院以及数据科学数学研究所，约翰霍普金斯大学）；Department of Statistics, Oxford-Man Institute of Quantitative Finance, and Nuffield College, University of Oxford（统计系、牛津-曼定量金融研究所以及牛津大学努尔菲尔德学院）；Management Science & Engineering Department, Stanford University（管理科学与工程系，斯坦福大学）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
We study singular perturbations of a class of two-scale stochastic control systems with unbounded data. The assumptions are designed to cover some relaxation problems for deep neural networks. We construct effective Hamiltonian and initial data and prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of probability, viscosity solutions and homogenization.
