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.

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.

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.

scholarly journals Conflict-Driven Heuristics for Mixed Integer Programming

2020 ◽  
Author(s):  
Jakob Witzig ◽  
Ambros Gleixner
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
State Of The Art ◽  
Computational Study ◽  
Search Tree ◽  
General Purpose ◽  
Conflict Analysis ◽  
Mixed Integer ◽  
Linear Programs ◽  
Feasible Solutions

Two essential ingredients of modern mixed-integer programming solvers are diving heuristics, which simulate a partial depth-first search in a branch-and-bound tree, and conflict analysis, which learns valid constraints from infeasible subproblems. So far, these techniques have mostly been studied independently: primal heuristics for finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. In this paper, we pose the question of whether and how the orthogonal goals of proving infeasibility and generating improving solutions can be pursued in a combined manner such that a state-of-the-art solver can benefit. To do so, we integrate both concepts in two different ways. First, we develop a diving heuristic that simultaneously targets the generation of valid conflict constraints from the Farkas dual and the generation of improving solutions. We show that, in the primal, this is equivalent to the optimistic strategy of diving toward the best bound with respect to the objective function. Second, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classic coefficient diving. In a detailed computational study, both methods are evaluated on the basis of an implementation in the source-open-solver SCIP. The experimental results underline the potential of combining both diving heuristics and conflict analysis. Summary of Contribution. This original article concerns the advancement of exact general-purpose algorithms for solving one of the largest and most prominent problem classes in optimization, mixed-integer linear programs. It demonstrates how methods for conflict analysis that learn from infeasible subproblems can be combined successfully with diving heuristics that aim at finding primal solutions. For two newly designed diving heuristics, this paper features a thoroughly computational study regarding their impact on the overall performance of a state-of-the-art MIP solver.

scholarly journals A Review of Deterministic Optimization Methods in Engineering and Management

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Hua Lin ◽  
Jung-Fa Tsai ◽  
Chian-Son Yu
Keyword(s):  
Nonlinear Programming ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Mixed Integer Nonlinear Programming ◽  
Mixed Integer ◽  
Deterministic Optimization ◽  
Practical Applications ◽  
Signomial Programming ◽  
Deterministic Methods ◽  
Integer Nonlinear Programming

With the increasing reliance on modeling optimization problems in practical applications, a number of theoretical and algorithmic contributions of optimization have been proposed. The approaches developed for treating optimization problems can be classified into deterministic and heuristic. This paper aims to introduce recent advances in deterministic methods for solving signomial programming problems and mixed-integer nonlinear programming problems. A number of important applications in engineering and management are also reviewed to reveal the usefulness of the optimization methods.

scholarly journals Conic reformulations for Kullback-Leibler divergence constrained distributionally robust optimization and applications

2021 ◽  
Vol 11 (2) ◽  
pp. 139-151
Author(s):  
Burak Kocuk
Keyword(s):  
Robust Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Empirical Distribution ◽  
Computational Study ◽  
Facility Location Problem ◽  
Optimization Approach ◽  
Distributionally Robust Optimization ◽  
Leibler Divergence ◽  
Distributionally Robust

In this paper, we consider a Kullback-Leibler divergence constrained distributionally robust optimization model. This model considers an ambiguity set that consists of all distributions whose Kullback-Leibler divergence to an empirical distribution is bounded. Utilizing the fact that this divergence measure has an exponential cone representation, we obtain the robust counterpart of the Kullback-Leibler divergence constrained distributionally robust optimization problem as a dual exponential cone constrained program under mild assumptions on the underlying optimization problem. The resulting conic reformulation of the original optimization problem can be directly solved by a commercial conic programming solver. We specialize our generic formulation to two classical optimization problems, namely, the Newsvendor Problem and the Uncapacitated Facility Location Problem. Our computational study in an out-of-sample analysis shows that the solutions obtained via the distributionally robust optimization approach yield significantly better performance in terms of the dispersion of the cost realizations while the central tendency deteriorates only slightly compared to the solutions obtained by stochastic programming.

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.

scholarly journals MIPLIB 2017: data-driven compilation of the 6th mixed-integer programming library

2021 ◽  
Author(s):  
Ambros Gleixner ◽  
Gregor Hendel ◽  
Gerald Gamrath ◽  
Tobias Achterberg ◽  
Michael Bastubbe ◽  
...  
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Optimization Problems ◽  
Selection Process ◽  
Data Driven ◽  
Mixed Integer ◽  
Integer Optimization ◽  
Programming Library ◽  
And Performance ◽  
First Time

AbstractWe report on the selection process leading to the sixth version of the Mixed Integer Programming Library, MIPLIB 2017. Selected from an initial pool of 5721 instances, the new MIPLIB 2017 collection consists of 1065 instances. A subset of 240 instances was specially selected for benchmarking solver performance. For the first time, these sets were compiled using a data-driven selection process supported by the solution of a sequence of mixed integer optimization problems, which encode requirements on diversity and balancedness with respect to instance features and performance data.

scholarly journals On decomposition and multiobjective-based column and disjunctive cut generation for MINLP

2020 ◽  
Author(s):  
Pavlo Muts ◽  
Ivo Nowak ◽  
Eligius M. T. Hendrix
Keyword(s):  
Nonlinear Programming ◽  
Optimization Problems ◽  
Mixed Integer ◽  
Resource Constrained ◽  
Linear Coupling ◽  
Integer Nonlinear Programming ◽  
Coupling Constraints ◽  
Cut Generation ◽  
Low Dimensional ◽  
Industrial Optimization

Abstract Most industrial optimization problems are sparse and can be formulated as block-separable mixed-integer nonlinear programming (MINLP) problems, defined by linking low-dimensional sub-problems by (linear) coupling constraints. This paper investigates the potential of using decomposition and a novel multiobjective-based column and cut generation approach for solving nonconvex block-separable MINLPs, based on the so-called resource-constrained reformulation. Based on this approach, two decomposition-based inner- and outer-refinement algorithms are presented and preliminary numerical results with nonconvex MINLP instances are reported.

scholarly journals A MILP Model for a Byzantine Fault Tolerant Blockchain Consensus

2020 ◽  
Vol 12 (11) ◽  
pp. 185
Author(s):  
Vitor Nazário Coelho ◽  
Rodolfo Pereira Araújo ◽  
Haroldo Gambini Santos ◽  
Wang Yong Qiang ◽  
Igor Machado Coelho
Keyword(s):  
Optimization Problems ◽  
Programming Model ◽  
Fault Tolerant ◽  
State Of The Art ◽  
Mixed Integer ◽  
Maximization Problem ◽  
Objective Functions ◽  
Objective Model ◽  
Adversarial Model ◽  
Algebraic Proofs

Mixed-integer mathematical programming has been widely used to model and solve challenging optimization problems. One interesting feature of this technique is the ability to prove the optimality of the achieved solution, for many practical scenarios where a linear programming model can be devised. This paper explores its use to model very strong Byzantine adversaries, in the context of distributed consensus systems. In particular, we apply the proposed technique to find challenging adversarial conditions on a state-of-the-art blockchain consensus: the Neo dBFT. Neo Blockchain has been using the dBFT algorithm since its foundation, but, due to the complexity of the algorithm, it is challenging to devise definitive algebraic proofs that guarantee safety/liveness of the system (and adjust for every change proposed by the community). Core developers have to manually devise and explore possible adversarial attacks scenarios as an exhaustive task. The proposed multi-objective model is intended to assist the search of possible faulty scenario, which includes three objective functions that can be combined as a maximization problem for testing one-block finality or a minimization problem for ensuring liveness. Automated graphics help developers to visually observe attack conditions and to quickly find a solution. This paper proposes an exact adversarial model that explores current limits for practical blockchain consensus applications such as dBFT, with ideas that can also be extended to other decentralized ledger technologies.

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