Modeling Defender-Attacker Problems as Robust Linear Programs with Mixed-Integer Uncertainty Sets

2021 ◽  
Author(s):  
Juan S. Borrero ◽  
Leonardo Lozano
Keyword(s):  
Robust Optimization ◽  
Convex Sets ◽  
Optimization Problems ◽  
Computational Study ◽  
Mixed Integer ◽  
Computational Time ◽  
Point Problem ◽  
Solution Algorithms ◽  
Uncertainty Sets ◽  
Uncertainty Set

We study a class of sequential defender-attacker optimization problems where the defender’s objective is uncertain and depends on the operations of the attacker, which are represented by a mixed-integer uncertainty set. The defender seeks to hedge against the worst possible data realization, resulting in a robust optimization problem with a mixed-integer uncertainty set, which requires the solution of a challenging mixed-integer problem, which can be seen as a saddle-point problem over a nonconvex domain. We study two exact solution algorithms and present two feature applications for which the uncertainty is naturally modeled as a mixed-integer set. Our computational experiments show that the considered algorithms greatly outperform standard algorithms both in terms of computational time and solution quality. Moreover, our results show that modeling uncertainty with mixed-integer sets, instead of approximating the data using convex sets, results in less conservative solutions, which translates to a lower cost for the defender to protect from uncertainty. Summary of Contribution: We consider a class of defender-attacker problems where the defender has to make operational decisions that depend on uncertain actions from an adversarial attacker. Due to the type of information available to the defender, neither probabilistic modeling, nor robust optimization methods with convex uncertainty sets, are well suited to address the defender’s decision-making problem. Consequently, we frame the defender’s problem as a class of robust optimization problems with a mixed-integer uncertainty sets, and devise two exact algorithms that solve this class of problems. A comprehensive computational study shows that for the considered applications, our algorithms improves the performance of existing robust optimization approaches that can be adapted to solve this class of problems. Moreover, we show how mixed-integer uncertainty sets can reduce the level of over-conservatism that is a known issue of robust optimization approaches.

Radius of Robust Feasibility for Mixed-Integer Problems

2021 ◽  
Author(s):  
Frauke Liers ◽  
Lars Schewe ◽  
Johannes Thürauf
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Continuous Optimization ◽  
Mixed Integer ◽  
Lp Relaxation ◽  
Maximal Size ◽  
Uncertainty Sets ◽  
Uncertainty Set ◽  
New Methodologies ◽  
Continuous Optimization Problems

For a mixed-integer linear problem (MIP) with uncertain constraints, the radius of robust feasibility (RRF) determines a value for the maximal size of the uncertainty set such that robust feasibility of the MIP can be guaranteed. The approaches for the RRF in the literature are restricted to continuous optimization problems. We first analyze relations between the RRF of a MIP and its continuous linear (LP) relaxation. In particular, we derive conditions under which a MIP and its LP relaxation have the same RRF. Afterward, we extend the notion of the RRF such that it can be applied to a large variety of optimization problems and uncertainty sets. In contrast to the setting commonly used in the literature, we consider for every constraint a potentially different uncertainty set that is not necessarily full-dimensional. Thus, we generalize the RRF to MIPs and to include safe variables and constraints; that is, where uncertainties do not affect certain variables or constraints. In the extended setting, we again analyze relations between the RRF for a MIP and its LP relaxation. Afterward, we present methods for computing the RRF of LPs and of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF. Summary of Contribution: Robust optimization is an important field of operations research due to its capability of protecting optimization problems from data uncertainties that are usually defined via so-called uncertainty sets. Intensive research has been conducted in developing algorithmically tractable reformulations of the usually semi-infinite robust optimization problems. However, in applications it also important to construct appropriate uncertainty sets (i.e., prohibiting too conservative, intractable, or even infeasible robust optimization problems due to the choice of the uncertainty set). In doing so, it is useful to know the maximal “size” of a given uncertainty set such that a robust feasible solution still exists. In this paper, we study one notion of “size”: the radius of robust feasibility (RRF). We contribute on the theoretical side by generalizing the RRF to MIPs as well as to include “safe” variables and constraints (i.e., where uncertainties do not affect certain variables or constraints). This allows to apply the RRF to many applications since safe variables and constraints exist in most applications. We also provide first methods for computing the RRF of LPs as well as of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF.

scholarly journals Data Driven Robust Energy and Reserve Dispatch Based on a Nonparametric Dirichlet Process Gaussian Mixture Model

Energies ◽  
2020 ◽  
Vol 13 (18) ◽  
pp. 4642
Author(s):  
Li Dai ◽  
Dahai You ◽  
Xianggen Yin
Keyword(s):  
Robust Optimization ◽  
Dirichlet Process ◽  
Forecast Error ◽  
Optimization Methods ◽  
Optimization Method ◽  
Gaussian Mixture ◽  
Data Driven ◽  
Uncertainty Sets ◽  
Uncertainty Set ◽  
Simulation Results

Traditional robust optimization methods use box uncertainty sets or gamma uncertainty sets to describe wind power uncertainty. However, these uncertainty sets fail to utilize wind forecast error probability information and assume that the wind forecast error is symmetrical and independent. This assumption is not reasonable and makes the optimization results conservative. To avoid such conservative results from traditional robust optimization methods, in this paper a novel data driven optimization method based on the nonparametric Dirichlet process Gaussian mixture model (DPGMM) was proposed to solve energy and reserve dispatch problems. First, we combined the DPGMM and variation inference algorithm to extract the GMM parameter information embedded within historical data. Based on the parameter information, a data driven polyhedral uncertainty set was proposed. After constructing the uncertainty set, we solved the robust energy and reserve problem. Finally, a column and constraint generation method was employed to solve the proposed data driven optimization method. We used real historical wind power forecast error data to test the performance of the proposed uncertainty set. The simulation results indicated that the proposed uncertainty set had a smaller volume than other data driven uncertainty sets with the same predefined coverage rate. Furthermore, the simulation was carried on PJM 5-bus and IEEE-118 bus systems to test the data driven optimization method. The simulation results demonstrated that the proposed optimization method was less conservative than traditional data driven robust optimization methods and distributionally robust optimization methods.

Data-Driven Optimization of Reward-Risk Ratio Measures

2020 ◽  
Author(s):  
Ran Ji ◽  
Miguel A. Lejeune
Keyword(s):  
Optimization Problems ◽  
State Of The Art ◽  
Convergence Property ◽  
Computational Study ◽  
Data Driven ◽  
Mixed Integer ◽  
Finite Convergence ◽  
Integer Nonlinear Programming ◽  
Distributionally Robust ◽  
Risk Ratios

We investigate a class of fractional distributionally robust optimization problems with uncertain probabilities. They consist in the maximization of ambiguous fractional functions representing reward-risk ratios and have a semi-infinite programming epigraphic formulation. We derive a new fully parameterized closed-form to compute a new bound on the size of the Wasserstein ambiguity ball. We design a data-driven reformulation and solution framework. The reformulation phase involves the derivation of the support function of the ambiguity set and the concave conjugate of the ratio function. We design modular bisection algorithms which enjoy the finite convergence property. This class of problems has wide applicability in finance, and we specify new ambiguous portfolio optimization models for the Sharpe and Omega ratios. The computational study shows the applicability and scalability of the framework to solve quickly large, industry-relevant-size problems, which cannot be solved in one day with state-of-the-art mixed-integer nonlinear programming (MINLP) solvers.

The cycle shop scheduling: mathematical model, properties and solution algorithms

2019 ◽  
Vol 53 (1) ◽  
pp. 111-128
Author(s):  
Bahman Naderia ◽  
Sheida Goharib
Keyword(s):  
High Performance ◽  
Scatter Search ◽  
Mixed Integer ◽  
Computational Time ◽  
Neighborhood Search ◽  
Scheduling Problems ◽  
Shop Scheduling ◽  
Solution Algorithms ◽  
Proposed Model ◽  
Constructive Heuristics

Conventionally, in scheduling problems it is assumed that each job visits each machine once. This paper studies a novel shop scheduling called cycle shop problems where jobs might return to each machine more than once. The problem is first formulated by two mixed integer linear programming models. The characteristics of the problem are analyzed, and it is realized that the problem suffers from a shortcoming called redundancy, i.e., several sequences represents the same schedule. In this regard, some properties are introduced by which the redundant sequences can be recognized before scheduling. Three constructive heuristics are developed. They are based on the shortest processing time first, insertion neighborhood search and non-delay schedules. Then, a metaheuristic based on scatter search is proposed. The algorithms are equipped with the redundancy prevention properties that greatly reduce the computational time of the algorithms. Two sets of experiments are conducted. The proposed model and algorithms are evaluated. The results show the high performance of model and algorithms.

scholarly journals Inverse Mixed Integer Optimization: Polyhedral Insights and Trust Region Methods

2022 ◽  
Author(s):  
Merve Bodur ◽  
Timothy C. Y. Chan ◽  
Ian Yihang Zhu
Keyword(s):  
Optimization Problems ◽  
Linear Optimization ◽  
Trust Region ◽  
Inverse Optimization ◽  
Mixed Integer ◽  
Integer Optimization ◽  
Solution Algorithms ◽  
Linear Optimization Problems ◽  
Mixed Integer Linear Optimization ◽  
Plane Solution

Inverse optimization—determining parameters of an optimization problem that render a given solution optimal—has received increasing attention in recent years. Although significant inverse optimization literature exists for convex optimization problems, there have been few advances for discrete problems, despite the ubiquity of applications that fundamentally rely on discrete decision making. In this paper, we present a new set of theoretical insights and algorithms for the general class of inverse mixed integer linear optimization problems. Specifically, a general characterization of optimality conditions is established and leveraged to design new cutting plane solution algorithms. Through an extensive set of computational experiments, we show that our methods provide substantial improvements over existing methods in solving the largest and most difficult instances to date.

scholarly journals K-adaptability in two-stage mixed-integer robust optimization

2019 ◽  
Vol 12 (2) ◽  
pp. 193-224 ◽  
Author(s):  
Anirudh Subramanyam ◽  
Chrysanthos E. Gounaris ◽  
Wolfram Wiesemann
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Mixed Integer ◽  
Finite Convergence ◽  
Two Stage ◽  
Infinite Dimensional ◽  
Benchmark Data ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation

Abstract We study two-stage robust optimization problems with mixed discrete-continuous decisions in both stages. Despite their broad range of applications, these problems pose two fundamental challenges: (i) they constitute infinite-dimensional problems that require a finite-dimensional approximation, and (ii) the presence of discrete recourse decisions typically prohibits duality-based solution schemes. We address the first challenge by studying a K-adaptability formulation that selects K candidate recourse policies before observing the realization of the uncertain parameters and that implements the best of these policies after the realization is known. We address the second challenge through a branch-and-bound scheme that enjoys asymptotic convergence in general and finite convergence under specific conditions. We illustrate the performance of our algorithm in numerical experiments involving benchmark data from several application domains.

Distributed Manufacturing Networks

2012 ◽  
Vol 4 (3) ◽  
pp. 25-42 ◽  
Author(s):  
Nathan Egge ◽  
Alexander Brodsky ◽  
Igor Griva
Keyword(s):  
Query Language ◽  
Optimization Problems ◽  
Static Problem ◽  
Decomposition Algorithm ◽  
Mixed Integer ◽  
Cost Functions ◽  
Computational Time ◽  
Finite Domain ◽  
Distributed Manufacturing ◽  
Manufacturing Network

The authors consider optimization problems expressed in Decision Guidance Query Language that may involve linear arithmetic constraints, as well as finite domain and binary variables. They focus on Distributed Manufacturing Network optimization problems in which only a part of the problem is dynamic, i.e., the demand for the output products in a manufacturing network, whereas the rest of the problem is static, i.e., the connectivity graph of the assembly processes and the cost functions of machines. The authors propose the Online Decomposition Algorithm based on offline preprocessing that optimizes each static problem component for discretized values of shared constraint variables, and approximate the optimal aggregated utility functions. The Online Decomposition Algorithm uses the pre-processed approximated aggregated cost functions to decompose the original problem into smaller problems, and utilizes search heuristics for the combinatorial part of the problem based on the pre-processed look-up tables. They also conduct an initial experimental evaluation which shows that the Online Decomposition Algorithm, as compared with Mixed Integer Linear Programming, provides an order of magnitude improvement in terms of both computational time and the quality of found solutions for a class of problems for which pre-processing is possible.

The Value of Randomized Solutions in Mixed-Integer Distributionally Robust Optimization Problems

2021 ◽  
Author(s):  
Erick Delage ◽  
Ahmed Saif
Keyword(s):  
Robust Optimization ◽  
Facility Location ◽  
Column Generation ◽  
Location Problem ◽  
Optimization Problems ◽  
Facility Location Problem ◽  
Mixed Integer ◽  
Distributionally Robust Optimization ◽  
Continuous Relaxation ◽  
Distributionally Robust

Randomized decision making refers to the process of making decisions randomly according to the outcome of an independent randomization device, such as a dice roll or a coin flip. The concept is unconventional, and somehow counterintuitive, in the domain of mathematical programming, in which deterministic decisions are usually sought even when the problem parameters are uncertain. However, it has recently been shown that using a randomized, rather than a deterministic, strategy in nonconvex distributionally robust optimization (DRO) problems can lead to improvements in their objective values. It is still unknown, though, what is the magnitude of improvement that can be attained through randomization or how to numerically find the optimal randomized strategy. In this paper, we study the value of randomization in mixed-integer DRO problems and show that it is bounded by the improvement achievable through its continuous relaxation. Furthermore, we identify conditions under which the bound is tight. We then develop algorithmic procedures, based on column generation, for solving both single- and two-stage linear DRO problems with randomization that can be used with both moment-based and Wasserstein ambiguity sets. Finally, we apply the proposed algorithm to solve three classical discrete DRO problems: the assignment problem, the uncapacitated facility location problem, and the capacitated facility location problem and report numerical results that show the quality of our bounds, the computational efficiency of the proposed solution method, and the magnitude of performance improvement achieved by randomized decisions. Summary of Contribution: In this paper, we present both theoretical results and algorithmic tools to identify optimal randomized strategies for discrete distributionally robust optimization (DRO) problems and evaluate the performance improvements that can be achieved when using them rather than classical deterministic strategies. On the theory side, we provide improvement bounds based on continuous relaxation and identify the conditions under which these bound are tight. On the algorithmic side, we propose a finitely convergent, two-layer, column-generation algorithm that iterates between identifying feasible solutions and finding extreme realizations of the uncertain parameter. The proposed algorithm was implemented to solve distributionally robust stochastic versions of three classical optimization problems and extensive numerical results are reported. The paper extends a previous, purely theoretical work of the first author on the idea of randomized strategies in nonconvex DRO problems by providing useful bounds and algorithms to solve this kind of problems.

An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

2021 ◽  
Vol 21 (4) ◽  
Author(s):  
Tingting Xia ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Computational Time ◽  
Pareto Solutions ◽  
Worst Case ◽  
Multi Objective ◽  
Pareto Points ◽  
Single Objective

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design. To find robust Pareto fronts, multi-objective robust optimization (MORO) methods with inner–outer optimization structures usually have high computational complexity, which is a critical issue. Generally, in design problems, robust Pareto solutions lie somewhere closer to nominal Pareto points compared with randomly initialized points. The searching process for robust solutions could be more efficient if starting from nominal Pareto points. We propose a new method sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points where MOOPs with uncertainties are solved in two stages. The deterministic optimization problem and robustness metric optimization are solved in the first stage, where nominal Pareto solutions and the robust-most solutions are identified, respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and robust-most points. The proposed SARPF method can reduce a significant amount of computational time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values, which can reduce computational efforts further. The global solvers, NSGA-II for multi-objective cases and genetic algorithm (GA) for single-objective cases, are used in corresponding optimization processes. Three examples with the comparison with results from the previous method are presented to demonstrate the applicability and efficiency of the proposed method.

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