Adjustable Regret for Continuous Control of Conservatism and Competitive Ratio Analysis

A major issue of the increasingly popular robust optimization is the tendency to produce overly conservative solutions. This paper proposes a new parameterized robust criterion to offer smooth control of conservatism without tampering with the uncertainty set. Unlike many other intractable criteria, its tractability is attained for common types of linear problems by reformulating them into traditional linear robust optimization problems. Many properties of it are studied to help analyze multistage robust optimization problems for closed-form solutions and give rise to a new approach to competitive ratio analysis. Finally, the new criterion is applied to the well-studied robust one-way trading problem to demonstrate its potential. A closed-form solution is obtained, which not only facilitates a numerical study of its smooth control of conservatism, but leads to a much simpler competitive ratio analysis.Comment: 29 pages, 2 figure

Minmax regret combinatorial optimization problems: an Algorithmic Perspective

Candia-Vejar, A (reprint author), Univ Talca, Modeling & Ind Management Dept, Curico, Chile.Uncertainty in optimization is not a new ingredient. Diverse models considering uncertainty have been developed over the last 40 years. In our paper we essentially discuss a particular uncertainty model associated with combinatorial optimization problems, developed in the 90's and broadly studied in the past years. This approach named minmax regret (in particular our emphasis is on the robust deviation criteria) is different from the classical approach for handling uncertainty, stochastic approach, where uncertainty is modeled by assumed probability distributions over the space of all possible scenarios and the objective is to find a solution with good probabilistic performance. In the minmax regret (MMR) approach, the set of all possible scenarios is described deterministically, and the search is for a solution that performs reasonably well for all scenarios, i.e., that has the best worst-case performance. In this paper we discuss the computational complexity of some classic combinatorial optimization problems using MMR. approach, analyze the design of several algorithms for these problems, suggest the study of some specific research problems in this attractive area, and also discuss some applications using this model