scholarly journals On Minimal Valid Inequalities for Mixed Integer Conic Programs

2016 ◽  
Vol 41 (2) ◽  
pp. 477-510 ◽  
Author(s):  
Fatma Kılınç-Karzan
Keyword(s):  
Valid Inequalities ◽  
Mixed Integer ◽  
Conic Programs

Multirow Intersection Cuts Based on the Infinity Norm

2021 ◽  
Author(s):  
Álinson S. Xavier ◽  
Ricardo Fukasawa ◽  
Laurent Poirrier
Keyword(s):  
Optimization Problems ◽  
Valid Inequalities ◽  
Linear Optimization ◽  
Computational Cost ◽  
General Purpose ◽  
Mixed Integer ◽  
Infinity Norm ◽  
Intersection Cuts ◽  
Linear Optimization Problems ◽  
Mixed Integer Linear Optimization

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.

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

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.

A New Formulation for the Dial-a-Ride Problem

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.

Mixed-integer nonlinear optimization for district heating network expansion

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.

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

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.

scholarly journals A multistage stochastic lot-sizing problem with cancellation and postponement under uncertain demands

2021 ◽  
Author(s):  
Carlos E Testuri ◽  
Héctor Cancela ◽  
Víctor M. Albornoz
Keyword(s):  
Information Structure ◽  
Lot Sizing ◽  
Valid Inequalities ◽  
Mixed Integer ◽  
Uncertain Demand ◽  
Programming Approach ◽  
Lead Times ◽  
Multi Stage ◽  
Lot Sizing Problem ◽  
Capacitated Lot Sizing

A multistage stochastic capacitated discrete procurement problem with lead times, cancellation and postponement is addressed.  The problem determines the procurement of a product under uncertain demand at minimal expected cost during a time horizon.  The supply of the product is made through the purchase of optional distinguishable orders of fixed size with delivery time.  Due to the unveiling of uncertainty over time it is possible to make cancellation and postponement corrective decisions on order procurement.  These decisions involve costs and times of implementation.  A model of the problem is formulated as an extension of a discrete capacitated lot-sizing problem under uncertain demand and lead times through a multi-stage stochastic mixed-integer linear programming approach.  Valid inequalities are generated, based on a conventional inequalities approach, to tighten the model formulation.  Experiments are performed for several problem instances with different uncertainty information structure.  Their results allow to conclude that the incorporation of a subset of the generated inequalities favor the model solution.

scholarly journals Integer Programming on the Junction Tree Polytope for Influence Diagrams

2020 ◽  
Vol 2 (3) ◽  
pp. 209-228
Author(s):  
Axel Parmentier ◽  
Victor Cohen ◽  
Vincent Leclère ◽  
Guillaume Obozinski ◽  
Joseph Salmon
Keyword(s):  
Graphical Model ◽  
Optimization Problems ◽  
Valid Inequalities ◽  
Random Variables ◽  
Mixed Integer ◽  
Influence Diagrams ◽  
Probabilistic Graphical Model ◽  
Computationally Efficient ◽  
Junction Tree ◽  
Acyclic Digraph

Influence diagrams (ID) and limited memory influence diagrams (LIMID) are flexible tools to represent discrete stochastic optimization problems, with the Markov decision process (MDP) and partially observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In an ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types: chance, decision, and utility vertices. The user chooses the distribution of the decision vertices conditionally to their parents in order to maximize the expected utility. Leveraging the notion of rooted junction tree, we present a mixed integer linear formulation for solving an ID, as well as valid inequalities, which lead to a computationally efficient algorithm. We also show that the linear relaxation yields an optimal integer solution for instances that can be solved by the “single policy update,” the default algorithm for addressing IDs.

scholarly journals Robust Mixed 0-1 Programming and Submodularity

2021 ◽  
pp. ijoo.2019.0042
Author(s):  
Seulgi Joung ◽  
Sungsoo Park
Keyword(s):  
Optimization Problems ◽  
Valid Inequalities ◽  
Continuous Variable ◽  
Integer Program ◽  
Data Uncertainty ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Robust Solution ◽  
Polyhedral Study ◽  
Set Function

The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.

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