Axis-Aligned Relaxations for Mixed-Integer Nonlinear Programming

arXiv (Cornell University) · 2026-03-19

We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.

Axis-Aligned Relaxations for Mixed-Integer Nonlinear Programming

ArXiv.org · 2026-03-19

We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.

New and Tighter Recovery Relations for Distillation Product Streams - Partial Reflux Case

SSRN Electronic Journal · 2026-01-01

New finite relaxation hierarchies for concavo-convex, disjoint bilinear programs, and facial disjunctions

Mathematical Programming · 2026-03-26

Abstract This paper introduces novel relaxation hierarchies for concavo-convex programs (CXP), a class of problems that includes disjoint bilinear programming (DBP) and concave minimization (CM) as special cases. At the core of these hierarchies is an algorithm based on double-description (DD) that computes the barycentric coordinates of a polyhedral cone as rational, non-negative functions representing multipliers associated with the cone’s rays. These hierarchies combine geometric structure derived from barycentric coordinates with algebraic techniques via rational functions, achieving the convex hull in m iterations, where m is the number of inequalities that a subset of the variables must satisfy. Our framework offers the first unified approach to analyze and tighten relaxations from disjunctive programming (DP) and reformulation-linearization technique (RLT) for CXP. We also demonstrate that our methods extend to facial disjunctive programs (FDP), where solutions are constrained to lie on faces of a Cartesian product of polytopes, generalizing known hierarchies for 0-1 programs.

Hattrick: Solving Multi-Class TE using Neural Models

2025-08-27

While recent work shows ML-based approaches are a promising alternative to conventional optimization methods for Traffic Engineering (TE), existing research is limited to a single traffic class. In this paper, we present Hattrick, the first ML-based approach for handling multiple traffic classes, a key requirement of cloud and ISP WANs. As part of Hattrick we have developed (i) a novel neural architecture aligned with the sequence of optimization problems in multiclass TE; and (ii) a variant of classical multitask learning methods to deal with the unique challenge of optimizing multiple metrics that have a precedence relationship. Evaluations on a large private WAN and other public datasets show Hattrick outperforms state-of-the-art optimization-based multiclass TE methods by better coping with prediction error - e.g., for GEANT, Hattrick outperforms SWAN by 5.48% to 19.3% across classes when considering the traffic that can be supported 99% of the time.

From Trees to Closed Loops: Inventory Management in Treewidth-Bounded Supply Chain Networks

SSRN Electronic Journal · 2025-01-01

Minimum reflux calculation for multicomponent distillation in multi‐feed, multi‐product columns: Algorithms and examples

AIChE Journal · 2025-08-11 · 1 citations

Abstract In this work, we present the first algorithm for identifying the minimum reboiler vapor duty requirement for a general multi‐feed, multi‐product (MFMP) distillation column separating ideal multicomponent mixtures. This algorithm incorporates our recently developed shortcut model for MFMP columns. We demonstrate the accuracy and efficiency of this algorithm through case studies. The results obtained from these case studies provide valuable insights into the optimal design of MFMP columns. Many of these insights go against the existing design guidelines and heuristics. For example, placing a colder saturated feed stream above a hotter saturated feed stream sometimes leads to a higher energy requirement. Furthermore, decomposing a general MFMP column into individual simple columns may lead to incorrect estimation of the minimum reflux ratio for the MFMP column. Thus, the algorithm presented here offers a fast, accurate, and automated approach to synthesize new, energy‐efficient, and cost‐effective MFMP columns.

Comparing exergy and heat pump work in distillation systems

AIChE Journal · 2025-05-21 · 1 citations

Abstract This study compares the exergy supplied to a distillation column with the actual energy consumption of a heat pump‐assisted distillation (HPAD) system. It evaluates the reliability of exergy as a proxy for heat pump work demand in such systems. While exergy can serve as a reasonable approximation when the thermal duties of the source and sink are nearly equal, it is not a consistent indicator of heat pump work in general. To address this limitation, we propose the Minimum Heat Pump Work Model, which establishes a theoretical lower bound on the work required to operate a heat pump between a given set of heat sources and sinks. Unlike exergy, this model is based on the direct thermal interaction between the source and sink, independent of the environment. It offers more accurate predictions of heat pump work and provides useful insights for designing energy‐efficient configurations in hybrid and multicomponent distillation systems.

Minimum reflux calculation for multicomponent distillation in multi-feed, multi-product columns: Algorithms and examples

arXiv (Cornell University) · 2025-03-26

In this work, we present the first algorithm for identifying the minimum reboiler vapor duty requirement for a general multi-feed, multi-product (MFMP) distillation column separating ideal multicomponent mixtures. This algorithm incorporates our latest advancement in developing the first shortcut model for MFMP columns. We demonstrate the accuracy and efficiency of this algorithm through case studies. The results obtained from these case studies also provide valuable insights on optimal design of MFMP columns. Many of these insights are against the existing design guidelines and heuristics. For example, placing a colder saturated feed stream above a hotter saturated feed stream sometimes leads to higher energy requirement. Furthermore, decomposing a general MFMP column into individual simple columns may lead to incorrect estimation of the minimum reflux ratio for the MFMP column. Thus, the algorithm presented here offers a fast, accurate, and automated approach to synthesize new, energy-efficient, and cost-effective MFMP columns.

A computationally efficient heat pump model for quick and reliable identification of energy-efficient distillation configurations

Chemical Engineering Science · 2025-10-01 · 1 citations