Valid Inequalities Based on Simple Mixed-Integer Sets

When generating multirow intersection cuts for mixed-integer linear optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet defining for the corner relaxation, the number of potential candidates is still very large, especially for instances of large size. In this paper, we introduce a subset of intersection cuts based on the infinity norm that is very small, works for relaxations having arbitrary number of rows and, unlike many subclasses studied in the literature, takes into account the entire data from the simplex tableau. We describe an algorithm for generating these inequalities and run extensive computational experiments in order to evaluate their practical effectiveness in real-world instances. We conclude that this subset of inequalities yields, in terms of gap closure, around 50% of the benefits of using all valid inequalities for the corner relaxation simultaneously, but at a small fraction of the computational cost, and with a very small number of cuts. Summary of Contribution: Cutting planes are one of the most important techniques used by modern mixed-integer linear programming solvers when solving a variety of challenging operations research problems. The paper advances the state of the art on general-purpose multirow intersection cuts by proposing a practical and computationally friendly method to generate them.

Download Full-text

Algorithms for the power-p Steiner tree problem in the Euclidean plane

Revista de Informática Teórica e Aplicada ◽

10.22456/2175-2745.80525 ◽

2018 ◽

Vol 25 (4) ◽

pp. 28

Author(s):

Christina Burt ◽

Alysson Costa ◽

Charl Ras

Keyword(s):

Spanning Tree ◽

Minimum Spanning Tree ◽

Valid Inequalities ◽

Minimum Cost ◽

Performance Ratio ◽

Mixed Integer ◽

Steiner Tree Problem ◽

Steiner Points ◽

Tree Construction ◽

The Cost

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

Download Full-text

A New Formulation for the Dial-a-Ride Problem

Transportation Science ◽

10.1287/trsc.2021.1044 ◽

2021 ◽

Author(s):

Yannik Rist ◽

Michael A. Forbes

Keyword(s):

Time Window ◽

Valid Inequalities ◽

Route Planning ◽

Branch And Cut ◽

Mixed Integer ◽

Planning Problem ◽

Current State ◽

New Formulation ◽

First Time ◽

Delivery Location

This paper proposes a new mixed integer programming formulation and branch and cut (BC) algorithm to solve the dial-a-ride problem (DARP). The DARP is a route-planning problem where several vehicles must serve a set of customers, each of which has a pickup and delivery location, and includes time window and ride time constraints. We develop “restricted fragments,” which are select segments of routes that can represent any DARP route. We show how to enumerate these restricted fragments and prove results on domination between them. The formulation we propose is solved with a BC algorithm, which includes new valid inequalities specific to our restricted fragment formulation. The algorithm is benchmarked on existing and new instances, solving nine existing instances to optimality for the first time. In comparison with current state-of-the-art methods, run times are reduced between one and two orders of magnitude on large instances.

Download Full-text

On Valid Inequalities for Mixed Integer p-Order Cone Programming

Journal of Optimization Theory and Applications ◽

10.1007/s10957-013-0315-7 ◽

2013 ◽

Vol 160 (2) ◽

pp. 439-456 ◽

Cited By ~ 3

Author(s):

Alexander Vinel ◽

Pavlo Krokhmal

Keyword(s):

Valid Inequalities ◽

Mixed Integer ◽

Cone Programming

Download Full-text

Valid inequalities for mixed-integer programmes with fixed charges on sets of variables

Operations Research Letters ◽

10.1016/j.orl.2020.03.004 ◽

2020 ◽

Vol 48 (3) ◽

pp. 240-244

Author(s):

Adam N. Letchford ◽

Georgia Souli

Keyword(s):

Valid Inequalities ◽

Mixed Integer ◽

Fixed Charges

Download Full-text

Mixed-integer nonlinear optimization for district heating network expansion

at - Automatisierungstechnik ◽

10.1515/auto-2020-0063 ◽

2020 ◽

Vol 68 (12) ◽

pp. 985-1000

Author(s):

Marius Roland ◽

Martin Schmidt

Keyword(s):

Nonlinear Optimization ◽

Optimization Model ◽

Thermal Effects ◽

Nonlinear Models ◽

Valid Inequalities ◽

Average Power ◽

District Heating ◽

Mixed Integer ◽

Mixed Integer Nonlinear Optimization

AbstractWe present a mixed-integer nonlinear optimization model for computing the optimal expansion of an existing tree-shaped district heating network given a number of potential new consumers. To this end, we state a stationary and nonlinear model of all hydraulic and thermal effects in the pipeline network as well as nonlinear models for consumers and the network’s depot. For the former, we consider the Euler momentum and the thermal energy equation. The thermal aspects are especially challenging. Here, we develop a novel polynomial approximation that we use in the optimization model. The expansion decisions are modeled by binary variables for which we derive additional valid inequalities that greatly help to solve the highly challenging problem. Finally, we present a case study in which we identify three major aspects that strongly influence investment decisions: the estimated average power demand of potentially new consumers, the distance between the existing network and the new consumers, and thermal losses in the network.

Download Full-text

A Hybrid IP/GA Approach to the Parallel Production Lines Scheduling Problem

Discrete Dynamics in Nature and Society ◽

10.1155/2016/5201937 ◽

2016 ◽

Vol 2016 ◽

pp. 1-11

Author(s):

Huizhi Ren ◽

Shenshen Sun

Keyword(s):

Linear Programming ◽

Integer Linear Programming ◽

Mixed Integer Linear Programming ◽

Time Window ◽

Valid Inequalities ◽

Mixed Integer ◽

Scheduling Problem ◽

Production Lines ◽

Solution Approach ◽

Cp Decomposition

A special parallel production lines scheduling problem is studied in this paper. Considering the time window and technical constraints, a mixed integer linear programming (MILP) model is formulated for the problem. A few valid inequalities are deduced and a hybrid mixed integer linear programming/constraint programming (MILP/CP) decomposition strategy is introduced. Based on them, a hybrid integer programming/genetic algorithm (IP/GA) approach is proposed to solve the problem. At last, the numerical experiments demonstrate that the proposed solution approach is effective and efficient.

Download Full-text