Repository – Page 26 – Optimization Online

The Convexity Zoo: A Taxonomy of Function Classes in Optimization

Published: 2026/01/01, Updated: 2026/05/10

Abbas Khademi

Generalized Convexity/Monoticity, Nonlinear Optimization, Optimization in Data Science generalized convexity, nonconvex optimization, quasar-convexity, star-convexity, structured nonconvexity

The tractability of optimization problems depends critically on structural properties of the objective function. Convexity guarantees global optimality of local solutions and enables polynomial-time algorithms under mild assumptions, but many problems arising in modern applications—particularly in machine learning—are inherently nonconvex. Remarkably, a large class of such problems remains amenable to efficient optimization due to additional … Read more

A Gauge Set Framework for Flexible Robustness Design

Published: 2025/12/31, Updated: 2026/01/22

Robust Optimization, Stochastic Programming

This paper proposes a unified framework for designing robustness in optimization under uncertainty using gauge sets, convex sets that generalize distance and capture how distributions may deviate from a nominal reference. Representing robustness through a gauge set reweighting formulation brings many classical robustness paradigms under a single convex-analytic perspective. The corresponding dual problem, the upper … Read more

Machine Learning–Enhanced Column Generation for Large-Scale Capacity Planning Problems

Published: 2025/12/29

(Mixed) Integer Linear Programming, Energy, Facility Planning and Design capacity planning, column generation, machine learning

Capacity Planning problems are a class of optimization problems used in diverse industries to improve resource allocation and make investment decisions. Solving real-world instances of these problems typically requires significant computational effort. To tackle this, we propose machine-learning-aided column generation methods for solving large-scale capacity planning problems. Our goal is to accelerate column generation by … Read more

AI for Enhancing Operations Research of Agriculture and Energy

Published: 2025/12/29

(Mixed) Integer Linear Programming, Applications - OR and Management Sciences, Global Optimization Agricultural Planning, energy systems, Learning-Enhanced Optimization, mixed-integer nonlinear programming, optimal power flow, reinforcement learning, stochastic optimization, unit commitment

This paper surveys optimization problems arising in agriculture, energy systems, and water-energy coordination from an operations research perspective. These problems are commonly formulated as integer nonlinear programs, mixed-integer nonlinear programs, or combinatorial set optimization models, characterized by nonlinear physical constraints, discrete decisions, and intertemporal coupling. Such structures pose significant computational challenges in large-scale and repeated-solution … Read more

New Location Science Models with Applications to UAV-Based Disaster Relief

Published: 2025/12/29

Convex and Nonsmooth Optimization, Facility Planning and Design convex analysis, Generalized Sylvester-Fermat-Torricelli models, location science, nonsmooth optimization, Unmanned aerial vehicle (UAV)

Natural and human-made disasters can cause severe devastation and claim thousands of lives worldwide. Therefore, developing efficient methods for disaster response and management is a critical task for relief teams. One of the most essential components of effective response is the rapid collection of information about affected areas, damages, and victims. More data translates into … Read more

Solving Stengle’s Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations

Published: 2025/12/29

Didier Henrion

Semi-definite Programming moments, orthogonal polynomials, polynomial optimization, sums-of-squares, trigonometric polynomials

We revisit Stengle’s classical univariate polynomial optimization example $\min 1-x^2$ s.t. $(1-x^2)^3\ge 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r\ge 3$ has the exact value $-1/r(r-2)$. For this we construct in rational arithmetic a dual polynomial sum-of-squares (SOS) certificate and a primal moment sequence … Read more

A Proximal-Gradient Method for Solving Regularized Optimization Problems with General Constraints

Published: 2025/12/27

Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization composite optimization, constrained optimization, nonconvex optimization, nonlinear optimization, proximal gradient method, regularization, worst-case iteration-complexity

We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step acceptance or rejection. Under various assumptions, we establish a worst-case iteration complexity result, prove that limit points are first-order KKT points, … Read more

An active-set method for box-constrained multiobjective optimization

Published: 2025/12/27, Updated: 2025/12/29

Multi-Criteria Optimization active set method, box-constrained problems, global convergence, multiobjective optimization

We propose an active-set algorithm for smooth multiobjective optimization problems subject to box constraints. The method works on one face of the feasible set at a time, treating it as a lower-dimensional region on which the problem simplifies. At each iteration, the algorithm decides whether to remain on the current face or to move to … Read more

Global optimization of low-rank polynomials

Published: 2025/12/27

Global Optimization, Semi-definite Programming

This work considers polynomial optimization problems where the objective admits a low-rank canonical polyadic tensor decomposition. We introduce LRPOP (low-rank polynomial optimization), a new hierarchy of semidefinite programming relaxations for which the size of the semidefinite blocks is determined by the canonical polyadic rank rather than the number of variables. As a result, LRPOP can … Read more

Primal-dual resampling for solution validation in convex stochastic programming

Published: 2025/12/26, Updated: 2026/04/01

Convex Optimization, Optimization in Data Science, Stochastic Programming statistical inference, stochastic optimization, stochastic programming

Suppose we wish to determine the quality of a candidate solution to a convex stochastic program in which the objective function is a statistical functional parameterized by the decision variable and known deterministic constraints may be present. Inspired by stopping criteria in primal-dual and interior-point methods, we develop cancellation theorems that characterize the convergence of … Read more