Robust Binary Linear Programming Under Implementation Uncertainty (original) (raw)

Mathematical Modelling of Robust Optimization for Integer Programming Problem

Journal of Institute of Science and Technology, 2015

Dealing with data uncertainty in mathematical programming has been recognized as a central problem in optimization for a long time. There are two methods that have been proposed to address data uncertainty over the years, namely stochastic programming and robust optimization. In this short paper, we present the brief review of mathematical models of integer programming problem and robust optimization approach to solve them. Robust Binary optimization is also briefly presented.

A branch and bound algorithm for robust binary optimization with budget uncertainty

Mathematical Programming Computation

Since its introduction in the early 2000s, robust optimization with budget uncertainty has received a lot of attention. This is due to the intuitive construction of the uncertainty sets and the existence of a compact robust reformulation for (mixed-integer) linear programs. However, despite its compactness, the reformulation performs poorly when solving robust integer problems due to its weak linear relaxation. To overcome the problems arising from the weak formulation, we propose a bilinear formulation for robust binary programming, which is as strong as theoretically possible. From this bilinear formulation, we derive strong linear formulations as well as structural properties for robust binary optimization problems, which we use within a tailored branch and bound algorithm. We test our algorithm’s performance together with other approaches from the literature on a diverse set of “robustified” real-world instances from the MIPLIB 2017. Our computational study, which is the first t...

A Branch & Bound Algorithm for Robust Binary Optimization with Budget Uncertainty

2021

Since its introduction in the early 2000s, robust optimization with budget uncertainty has received a lot of attention. This is due to the intuitive construction of the uncertainty sets and the existence of a compact robust reformulation for (mixed-integer) linear programs. However, despite its compactness, the reformulation performs poorly when solving robust integer problems due to its weak linear relaxation. To overcome the problems arising from the weak formulation, we propose a bilinear formulation for robust binary programming, which is as strong as theoretically possible. From this bilinear formulation, we derive strong linear formulations as well as structural properties for robust binary optimization problems, which we use within a tailored branch & bound algorithm. We test our algorithm’s performance together with other approaches from the literature on a diverse set of “robustified” real-world instances from the MIPLIB 2017. Our computational study, which is the first to ...

Adjustable robust solutions of uncertain linear programs

Mathematical Programming, 2004

We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time. If the vector c is also uncertain, we could look at the equivalent formulation of (1): min x,t t : c T x ≤ t, Ax ≥ b and thus without loss of generality we may restrict the uncertainty to the constraints only.

Robust solutions of uncertain linear programs

Operations Research Letters, 1999

Robust solutions of Linear Programming problems contaminated with uncertain data

Mathematical Programming, 2000

Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed. We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization methodology (Ben-Tal and Nemirovski [1-3]; El Ghaoui et al. ) to produce "robust" solutions of the above LPs which are in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in optimality.

Robust solutions to multi-objective linear programs with uncertain data

European Journal of Operational Research, 2015

In this paper we examine multi-objective linear programming problems in the face of data uncertainty both in the objective function and the constraints. First, we derive a formula for the radius of robust feasibility guaranteeing constraint feasibility for all possible scenarios within a speci…ed uncertainty set under a¢ ne data parametrization. We then present numerically tractable optimality conditions for minmax robust weakly e¢ cient solutions, i.e., the weakly e¢ cient solutions of the robust counterpart. We also consider highly robust weakly e¢cient solutions, i.e., robust feasible solutions which are weakly e¢ cient for any possible instance of the objective matrix within a speci…ed uncertainty set, providing lower bounds for the radius of highly robust e¢ ciency guaranteeing the existence of this type of solutions under a¢ ne and rank-1 objective data uncertainty. Finally, we provide numerically tractable optimality conditions for highly robust weakly e¢ cient solutions.

Chapter One. Uncertain Linear Optimization Problems and their Robust Counterparts

2009

In this chapter, we introduce the concept of the uncertain Linear Optimization problem and its Robust Counterpart, and study the computational issues associated with the emerging optimization problems. 1.1 DATA UNCERTAINTY IN LINEAR OPTIMIZATION Recall that the Linear Optimization (LO) problem is of the form min x c T x + d : Ax ≤ b , (1.1.1) where x ∈ R n is the vector of decision variables, c ∈ R n and d ∈ R form the objective, A is an m × n constraint matrix, and b ∈ R m is the right hand side vector. Clearly, the constant term d in the objective, while affecting the optimal value, does not affect the optimal solution, this is why it is traditionally skipped. As we shall see, when treating the LO problems with uncertain data there are good reasons not to neglect this constant term. The structure of problem (1.1.1) is given by the number m of constraints and the number n of variables, while the data of the problem are the collection (c, d, A, b), which we will arrange into an (m + 1) × (n + 1) data matrix D = c T d A b. Usually not all constraints of an LO program, as it arises in applications, are of the form a T x ≤ const; there can be linear "≥" inequalities and linear equalities as well. Clearly, the constraints of the latter two types can be represented equivalently by linear "≤" inequalities, and we will assume henceforth that these are the only constraints in the problem. 1.1.1 Introductory Example Consider the following very simple linear optimization problem: Example 1.1.1. A company produces two kinds of drugs, DrugI and DrugII, containing a specific active agent A, which is extracted from raw materials purchased on the market. There are two kinds of raw materials, RawI and RawII, which can be used as sources of the active agent. The related production, cost, and resource data are given in table 1.1. The goal is to find the production plan that maximizes the profit of the company.

Decomposition approaches for two-stage robust binary optimization

Le Centre pour la Communication Scientifique Directe - HAL - Diderot, 2019

Introduction Theoretical development Application to robust capital budgeting Numerical results Conclusion 1 Introduction 2 Theoretical development 3 Application to robust capital budgeting 4 Numerical results 5 Conclusion Arslan and Detienne 2 / 31 Outline Introduction Theoretical development Application to robust capital budgeting Numerical results Conclusion Outline 1 Introduction 2 Theoretical development 3 Application to robust capital budgeting 4 Numerical results 5 Conclusion Arslan and Detienne 3 / 31 Outline Introduction Theoretical development Application to robust capital budgeting Numerical results Conclusion Static vs Two-stage robust optimization Static model : Decide x knowing only ξ ∈ Ξ Undertake decision x t Two-stage model : Decide x knowing only ξ ∈ Ξ t Decide recourse y (x, ξ) Actual outcome ξ revealed Arslan and Detienne 4 / 31 q(ξ), T (ξ), W (ξ), h(ξ) : uncertainty revealed after first-stage decisions are taken Arslan and Detienne 5 / 31 (i) Decision rules: Recourse decisions are restricted to be functions of uncertainty:

Optimization under Decision-Dependent Uncertainty

SIAM Journal on Optimization, 2018

The efficacy of robust optimization spans a variety of settings with uncertainties bounded in predetermined sets. In many applications, uncertainties are affected by decisions and cannot be modeled with current frameworks. This paper takes a step towards generalizing robust linear optimization to problems with decisiondependent uncertainties. In general settings, we show these problems to be NP-complete. To alleviate the computational inefficiencies, we introduce a class of uncertainty sets whose size depends on binary decisions. We propose reformulations that improve upon alternative standard linearization techniques. To illustrate the advantages of this framework, a shortest path problem is discussed, where the uncertain arc lengths are affected by decisions. Beyond the modeling and performance advantages, the proposed notion of proactive uncertainty control also mitigates over conservatism of current robust optimization approaches.

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