Finding Total Unimodularity in Optimization Problems Solved by Linear Programs

The paper proposes a novel approach for the interval limit analysis of rigid-perfectly plastic structures with (nonprobabilistic) uncertain but bounded forces and yield capacities that vary within given continuous ranges. The discrete model is constructed within a polygon-scaled boundary finite element framework, which advantageously provides coarse mesh accuracy even in the presence of stress singularities and complex geometry. The interval analysis proposed is based on a so-called convex model for the direct determination of both maximum and minimum collapse load limits of the structures involved. The formulation for this interval limit analysis takes the form of a pair of optimization problems, known as linear programs with interval coefficients (LPICs). This paper proposes a robust and efficient reformulation of the original LPICs into standard nonlinear programming (NLP) problems with bounded constraints that can be solved using any NLP code. The proposed NLP approach can capture, within a single step, the maximum collapse load limit in one case and the minimum collapse load limit in the other, and thus eliminates the need for any combinatorial search schemes.

Download Full-text

Solving Integer Linear Programs by Exploiting Variable-Constraint Interactions: A Survey

Algorithms ◽

10.3390/a12120248 ◽

2019 ◽

Vol 12 (12) ◽

pp. 248

Author(s):

Robert Ganian ◽

Sebastian Ordyniak

Keyword(s):

Linear Programming ◽

Computer Science ◽

Integer Linear Programming ◽

Systematic Study ◽

Optimization Problems ◽

Linear Programs ◽

Integer Linear Programs ◽

Diverse Backgrounds ◽

Recent Developments ◽

Np Complete

Integer Linear Programming (ILP) is among the most successful and general paradigms for solving computationally intractable optimization problems in computer science. ILP is NP-complete, and until recently we have lacked a systematic study of the complexity of ILP through the lens of variable-constraint interactions. This changed drastically in recent years thanks to a series of results that together lay out a detailed complexity landscape for the problem centered around the structure of graphical representations of instances. The aim of this survey is to summarize these recent developments, put them into context and a unified format, and make them more approachable for experts from many diverse backgrounds.

Download Full-text

Network-Based Approximate Linear Programming for Discrete Optimization

Operations Research ◽

10.1287/opre.2019.1953 ◽

2020 ◽

Vol 68 (6) ◽

pp. 1767-1786

Author(s):

Selvaprabu Nadarajah ◽

Andre A. Cire

Keyword(s):

Linear Programming ◽

Discrete Optimization ◽

Optimization Problems ◽

Linear Programs ◽

Solution Quality ◽

Marketing Analytics ◽

Dynamic Programs ◽

Approximate Linear Programming ◽

Commercial Solver ◽

Discrete Optimization Problems

Several prescriptive tasks in business and engineering as well as prediction in machine learning entail the solution of challenging discrete optimization problems. We recast the typical optimization formulation of these problems as high-dimensional dynamic programs and approach their approximation via linear programming. We develop tractable approximate linear programs with supporting theory by bringing together tools from state-space aggregations, networks, and perfect graphs (i.e., graph completions). We embed these models in a simple branch-and-bound scheme to solve applications in marketing analytics and the maintenance of energy or city-owned assets. We find that the resulting technique substantially outperforms a state-of-the-art commercial solver as well as aggregation-heuristics in terms of both solution quality and time. Our results motivate further consideration of networks and graph theory in approximate linear programming for solving deterministic and stochastic discrete optimization problems.

Download Full-text

An Active Set Sequential Linearization Algorithm for Nonlinear Design Optimization

13th Design Automation Conference: Volume 1 — Design Methods, Computer Graphics, and Expert Systems ◽

10.1115/detc1987-0001 ◽

1987 ◽

Author(s):

N. Tzannetakis ◽

P. Y. Papalambros

Keyword(s):

Design Optimization ◽

Optimization Problems ◽

Programming Algorithm ◽

Linear Programs ◽

Chemical Process Design ◽

Active Set ◽

Active Set Strategy ◽

Nonlinear Design ◽

Working Set ◽

Sequential Linearization

Abstract Solution of nonlinear design optimization problems via a sequence of linear programs is regaining attention for solving certain model classes, such as in structural design and chemical process design. An active set strategy modification of an algorithm by Palacios-Gomez is presented. A special interior linear programming algorithm with active set strategy is used also for solving the subproblem and generating the working set of the outer iterations. Examples are included.

Download Full-text

Balanced Ranking with Diversity Constraints

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2019/836 ◽

2019 ◽

Author(s):

Ke Yang ◽

Vasilis Gkatzelis ◽

Julia Stoyanovich

Keyword(s):

Optimization Problems ◽

Unintended Consequence ◽

Linear Programs ◽

Disadvantaged Populations ◽

Ranking Algorithms ◽

Integer Linear Programs ◽

Trade Offs ◽

Overall Performance ◽

Additional Constraints ◽

Selection And Ranking

Many set selection and ranking algorithms have recently been enhanced with diversity constraints that aim to explicitly increase representation of historically disadvantaged populations, or to improve the over-all representativeness of the selected set. An unintended consequence of these constraints, however, is reduced in-group fairness: the selected candidates from a given group may not be the best ones, and this unfairness may not be well-balanced across groups. In this paper we study this phenomenon using datasets that comprise multiple sensitive attributes. We then introduce additional constraints, aimed at balancing the in-group fairness across groups, and formalize the induced optimization problems as integer linear programs. Using these programs, we conduct an experimental evaluation with real datasets, and quantify the feasible trade-offs between balance and overall performance in the presence of diversity constraints.

Download Full-text

Some Zero-One Linear Programming Reformulations for the Maximum Clique Problem

Mathematica Pannonica ◽

10.1556/314.2020.00005 ◽

2021 ◽

Vol 27_NS1 (1) ◽

pp. 32-47

Author(s):

Ákos Beke ◽

Sándor Szabó ◽

Bogdán Zavalnij

Keyword(s):

Linear Programming ◽

Optimization Problems ◽

Maximum Clique ◽

Independent Set ◽

Maximum Clique Problem ◽

Linear Programs ◽

Combinatorial Optimization Problems ◽

Edge Covering ◽

Covering Set ◽

Clique Problem

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

Download Full-text

Efficiency of MOMA-plus Method to Solve Some Fully Fuzzy L-R Triangular Multiobjective Linear Programs

Journal of Mathematics Research ◽

10.5539/jmr.v10n2p77 ◽

2018 ◽

Vol 10 (2) ◽

pp. 77 ◽

Cited By ~ 1

Author(s):

Abdoulaye Compaoré ◽

Kounhinir Somé ◽

Joseph Poda ◽

Blaise Somé

Keyword(s):

Optimization Problem ◽

Optimization Problems ◽

Linear Optimization ◽

Linear Programs ◽

Pareto Optimal Solutions ◽

Optimal Solutions ◽

Linear Optimization Problem ◽

Novel Approach ◽

Linear Optimization Problems ◽

Single Objective

In this paper, we propose a novel approach for solving some fully fuzzy L-R triangular multiobjective linear optimization programs using MOMA-plus method (Kounhinir, 2017). This approach is composed of two relevant steps such as the converting of the fully fuzzy L-R triangular multiobjective linear optimization problem into a deterministic multiobjective linear optimization and the applying of the adapting MOMA-plus method. The initial version of MOMA-plus method is designed for multiobjective deterministic optimization (Kounhinir, 2017) and having already been tested on the single-objective fuzzy programs (Abdoulaye, 2017). Our new method allow to find all of the Pareto optimal solutions of a fully fuzzy L-R triangular multiobjective linear optimization problems obtained after conversion. For highlighting the efficiency of our approach a didactic numerical example is dealt with and obtained solutions are compared to Total Objective Segregation Method proposed by Jayalakslmi and Pandia (Jayalakslmi 2014).

Download Full-text