mixed-integer nonlinear programming

Modeling Binary Relations in Piecewise-Linear Approximations

Published: 2026/02/28, Updated: 2026/03/02

Combinatorial Optimization, Global Optimization Theory, Nonlinear Optimization Bipartite Implication Polytope, boolean quadric polytope, cutting planes, mixed-integer nonlinear programming, Multipartite Implication Polytope, piecewise-linear approximations

Over the last decades, using piecewise-linear mixed-integer relaxations of nonlinear expressions has become a strong alternative to spatial branching for solving mixed-integer nonlinear programs. Since these relaxations give rise to large numbers of binary variables that encode interval selections, strengthening them is crucial. We investigate how to exploit the resulting combinatorial structure by integrating cutting-plane … Read more

Fast Presolving Framework For Sparsity Constrained Convex Quadratic Programming: Screening-Based Cut Generation and Selection

Published: 2026/02/14

Haozhe Tan

Guanyi Wang

(Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Nonlinear Optimization Cutting plane, mixed-integer nonlinear programming, presolving, Screening, sparsity

Screening is widely utilized for Mixed-Integer Programming (MIP) presolving. It aims to certify a priori whether one or multiple specific binary variables can be fixed to optimal values based on solutions from convex relaxations. This paper studies the challenge of solving Sparsity-constrained (strongly) Convex Quadratic Programming (SCQP) and proposes the Screening-based Cut Presolving Framework (SCPF). … Read more

AI for Enhancing Operations Research of Agriculture and Energy

Published: 2025/12/29

Morteza Kimiaei

Anshu Kumar

Vyacheslav Kungurtsev

(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

Semidefinite programming via Projective Cutting Planes for dense (easily-feasible) instances

Published: 2025/12/01

Daniel Porumbel

Linear, Cone and Semidefinite Programming, Semi-definite Programming cutting planes, mixed-integer nonlinear programming, semidefinite programming

The cone of positive semi-definite (SDP) matrices can be described by an infinite number of linear constraints. It is well-known that one can optimize over such a feasible area by standard Cutting Planes, but work on this idea remains a rare sight, likely due to its limited practical appeal compared to Interior Point Methods (IPMs). … Read more

The SCIP Optimization Suite 10.0

Published: 2025/11/24

Integer Programming, Linear, Cone and Semidefinite Programming, Optimization Software and Modeling Systems branch-and-cut algorithm, Branch-and-price algorithm, column generation, constraint integer programming, linear programming, mixed-integer linear programming, mixed-integer nonlinear programming, mixed-integer semidefinite programming, optimization solver, parallelization

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming (CIP) framework SCIP. This report discusses the enhancements and extensions included in SCIP Optimization Suite 10.0. The updates in SCIP 10.0 include a new solving mode for exactly solving rational mixed-integer linear programs, a new presolver … Read more

Randomized Roundings for a Mixed-Integer Elliptic Control System

Published: 2024/09/12, Updated: 2025/09/04

Martin T. Köhler

Lauri Kröger

Christian Kirches

(Mixed) Integer Nonlinear Programming, Systems governed by Differential Equations Optimization mixed-integer nonlinear programming, pde constrained optimization problem, randomized algorithms

We present randomized reconstruction approaches for optimal solutions to mixed-integer elliptic PDE control systems. Approximation properties and relations to sum-up rounding are derived using the cut norm. This enables us to dispose of space-filling curves required for sum-up rounding. Rates of almost sure convergence in the cut norm and the SUR norm in control space … Read more

Granularity for mixed-integer polynomial optimization problems

Published: 2024/07/17, Updated: 2025/03/21

Carl Eggen

Oliver Stein

Stefan Volkwein

(Mixed) Integer Nonlinear Programming, Semi-definite Programming granularity, mixed-integer nonlinear programming, polynomial optimization, rounding, semidefinite programming

Finding good feasible points is crucial in mixed-integer programming. For this purpose we combine a sufficient condition for consistency, called granularity, with the moment-/sos-hierarchy from polynomial optimization. If the mixed-integer problem is granular, we obtain feasible points by solving continuous polynomial problems and rounding their optimal points. The moment-/sos-hierarchy is hereby used to solve those … Read more

An outer approximation method for solving mixed-integer convex quadratic programs with indicators

Published: 2023/12/08

Linchuan Wei

Simge Küçükyavuz

(Mixed) Integer Nonlinear Programming, Quadratic Programming convex analysis, mixed-integer nonlinear programming, quadratic programming

Mixed-integer convex quadratic programs with indicator variables (MIQP) encompass a wide range of applications, from statistical learning to energy, finance, and logistics. The outer approximation (OA) algorithm has been proven efficient in solving MIQP, and the key to the success of an OA algorithm is the strength of the cutting planes employed. In this paper, … Read more

Benchmarking Piecewise Linear Reformulations for MINLPs: A Computational Study Based on the Open-Source Framework PWL-T-Rex

Published: 2023/09/29, Updated: 2025/04/11

Kristin Braun

Robert Burlacu

(Mixed) Integer Nonlinear Programming discrete optimization, mixed-integer nonlinear programming, mixed-integer programming, piecewise linear relaxations

Solving mixed-integer nonlinear problems by means of piecewise linear relaxations has emerged as a reasonable alternative to the commonly used spatial branch-and-bound. These relaxations have been modeled by various mixed-integer models in recent decades. The idea is to exploit the availability of mature solvers for mixed-integer problems. In this work, we implement a framework that … Read more

Global Optimization of Mixed-Integer Nonlinear Programs with SCIP 8.0

Published: 2022/12/16

(Mixed) Integer Nonlinear Programming, Global Optimization, Optimization Software and Modeling Systems branch-and-cut, global optimization, mixed-integer nonlinear programming, scip

For over ten years, the constraint integer programming framework SCIP has been extended by capabilities for the solution of convex and nonconvex mixed-integer nonlinear programs (MINLPs). With the recently published version 8.0, these capabilities have been largely reworked and extended. This paper discusses the motivations for recent changes and provides an overview of features that … Read more