An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function

Categories Data Science Algorithms, Optimization in Data Science, Other Topics Tags , ,

Nonlinear matrix decomposition (NMD) with the ReLU function, denoted ReLU-NMD, is the following problem: given a sparse, nonnegative matrix \(X\) and a factorization rank \(r\), identify a rank-\(r\) matrix \(\Theta\) such that \(X\approx \max(0,\Theta)\). This decomposition finds application in data compression, matrix completion with entries missing not at random, and manifold learning. The standard ReLU-NMD … Read more

NonOpt: Nonconvex, Nonsmooth Optimizer

Categories Nonlinear Optimization, Nonsmooth Optimization, Unconstrained Optimization Tags , , , ,

NonOpt, a C++ software package for minimizing locally Lipschitz objective functions, is presented. The software is intended primarily for minimizing objective functions that are nonconvex and/or nonsmooth. The package has implementations of two main algorithmic strategies: a gradient-sampling and a proximal-bundle method. Each algorithmic strategy can employ quasi-Newton techniques for accelerating convergence in practice. The … Read more

Sequential Rounding in Mixed-Integer Model Predictive Control

Categories Approximation Algorithms, Control Applications, Systems governed by Differential Equations Optimization Tags , , ,

This paper interfaces combinatorial integral approximation strategies with the inherent robustness properties of conventional model predictive control with stabilizing terminal conditions to establish practical asymptotic stability results for finite control set and mixed-integer model predictive control. We examine the impact of sequential control rounding on the closed-loop performance in terms of stability and optimality. Sum-up … Read more

Local Linear Convergence of the Alternating Direction Method of Multipliers for Semidefinite Programming under Strict Complementarity

Categories Convex Optimization, Semi-definite Programming

We investigate the local linear convergence properties of the Alternating Direction Method of Multipliers (ADMM) when applied to Semidefinite Programming (SDP). A longstanding belief suggests that ADMM is only capable of solving SDPs to moderate accuracy, primarily due to its sublinear worst-case complexity and empirical observations of slow convergence. We challenge this notion by introducing … Read more

The Least Singular Value Function in Variational Analysis

Categories Constrained Nonlinear Optimization, Global Optimization Theory, Nonsmooth Optimization Tags , , , ,

Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, met- ric regularity can be elusive for some important ill-posed classes of problems includ- ing polynomial equations, parametric variational systems, smooth reformulations of complementarity systems with degenerate solutions, etc. The study of stability issues for such … Read more

Climate-Resilient Nodal Power System Expansion Planning for a Realistic California Test Case

Categories (Mixed) Integer Linear Programming, Energy, Stochastic Programming Tags , , , ,

Climate change is increasingly impacting power system operations, not only through more frequent extreme weather events but also through shifts in routine weather patterns. Factors such as increased temperatures, droughts, changing wind patterns, and solar irradiance shifts can impact both power system production and transmission and electric load. The current power system was not designed … Read more

Optimal Control of Semilinear Graphon Systems

Categories Infinite Dimensional Optimization, Network Optimization, Systems governed by Differential Equations Optimization Tags , , ,

We investigate optimal control of semilinear dynamical systems on asymptotically infinite networks, using the notion of graphons. A graphon represents a limit object of a converging graph sequence and serves as a generalization of a finite graph, enabling a systematic analysis of large-scale networks. We analyze semilinear graphon dynamical systems and establish state convergence along … Read more

New Sufficient and Necessary Conditions for Constrained and Unconstrained Lipschitzian Error Bounds

Categories Complementarity and Variational Inequalities, Global Optimization Theory, Nonsmooth Optimization Tags , , , , , , ,

Local error bounds play a fundamental role in mathematical programming and variational analysis. They are used e.g. as constraint qualifications in optimization, in developing calculus rules for generalized derivatives in nonsmooth and set-valued analysis, and they serve as a key ingredient in the design and convergence analysis of Newton-type methods for solving systems of possibly … Read more

Decision space decomposition for multiobjective optimization

Categories Multi-Criteria Optimization Tags , , , ,

Being inspired by the parametric decomposition theorem for multiobjective optimization problems (MOPs) of Cuenca Mira and Miguel García (2017), and by the block-coordinate descent for single objective optimization problems, we present a decomposition theorem for computing the set of minimal elements of a partially ordered set. This set is decomposed into subsets whose minimal elements … Read more

A constraint programming model and a hybrid iterated local search algorithm for solving an aircraft recovery problem in the oil and gas industry

Categories Applications - OR and Management Sciences, Other Topics Tags , , , ,

In this paper, we address a challenging problem faced by a Brazilian oil and gas company regarding the rescheduling of helicopter flights from an onshore airport to maritime units, crucial for transporting company employees. The problem arises due to unforeseen events like bad weather or mechanical failures, leading to delays or postponements in the original … Read more