Krzysztof Postek

Multi-Stage Adjustable Robust Mixed-Integer Optimization via Iterative Splitting of the Uncertainty Set

In this paper we propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problem finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets, one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem’s computational complexity stays at the same level, as for the static robust problem. This also holds in the nonfixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results on splitting the uncertainty set by identifying sets of uncertain parameter scenarios to be divided for an improvement in the worst-case objective value. Based on this theory, we propose several splitting heuristics. Numerical examples entailing...

Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures

In this paper we study distributionally robust constraints on risk measures (such as standard deviation less the mean, Conditional Value-at-Risk, Entropic Value-at-Risk) of decision-dependent random variables. The uncertainty sets for the discrete probability distributions are defined using statistical goodness-of-fit tests and probability metrics such as Pearson, likelihood ratio, Anderson-Darling tests, or Wasserstein distance. This type of constraints arises in problems in portfolio optimization, economics, machine learning, and engineering. We show that the derivation of a tractable robust counterpart can be split into two parts: one corresponding to the risk measure and the other to the uncertainty set. We also show how the counterpart can be constructed for risk measures that are nonlinear in the probabilities (for example, variance or the Conditional Value-at-Risk). We provide the computational tractability status for each of the uncertainty set-risk measure pairs that we could solve. Numerical examples including portfolio optimization and a multi-item newsvendor problem illustrate the proposed approach.

Exact Robust Counterparts of Ambiguous Stochastic Constraints Under Mean and Dispersion Information

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the old result of Ben-Tal and Hochman (1972) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. We use these bounds to derive exact robust counterparts of expected feasibility of convex constraints and to construct new safe tractable approximations of chance constraints. Numerical examples show our method to be applicable to numerous applications of Robust Optimization, e.g., where implementation error or linear decision rules are present. Also, we show that the methodology can be used for optimization the average-case performance of worst-case optimal Robust Optimization solutions.

Robust Optimization with Ambiguous Stochastic Constraints Under Mean and Dispersion Information

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the 1972 result of Ben-Tal and Hochman (BH) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. First, we use these results to treat ambiguous expected feasibility constraints to obtain exact reformulations for both functions that are convex and concave in the components of the random variable. This approach requires, however, the independence of the random variables and, moreover, may lead to an exponential number of terms in the resulting robust counterparts. We then show how upper bounds can be constructed that alleviate the independence restriction, and require only a linear number of terms, by exploiting models in which random variables are linearly aggregated. Moreover, using the BH bounds we derive three new safe tractable approximations of chance constraints of increasing computational complexity and quality. In a numerical study, we demonstrate the efficiency of our methods in solving stochastic optimization problems under mean-MAD ambiguity.

An approximation framework for two-stage ambiguous stochastic integer programs under mean-MAD information

Abstract We consider two-stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model. If all decision variables are continuous, then we are able to derive the worst-case and best-case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known. Contrary to most existing results in the literature, these probability distributions are the same for every first-stage decision. The ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables. For such problems, we develop a general approximation framework and derive error bounds for using these approximatons. We apply this approximation framework to mixed-ambiguous mixed-integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous. To illustrate these results we carry out numerical experiments on a surgery block allocation problem.

Efficient Methods for Several Classes of Ambiguous Stochastic Programming Problems Under Mean-MAD Information

We consider decision making problems under uncertainty, assuming that only partial distributional information is available - as is often the case in practice. In such problems, the goal is to determine here-and-now decisions, which optimally balance deterministic immediate costs and worst-case expected future costs. These problems are challenging, since the worst-case distribution needs to be determined while the underlying problem is already a difficult multistage recourse problem. Moreover, as found in many applications, the model may contain integer variables in some or all stages. Applying a well-known result by Ben-Tal and Hochman (1972), we are able to efficiently solve such problems without integer variables, assuming only readily available distributional information on means and mean-absolute deviations. Moreover, we extend the result to the non-convex integer setting by means of convex approximations (see Romeijnders et al. (2016a)), proving corresponding performance bounds. Our approach is straightforward to implement using of-the-shelf software as illustrated in our numerical experiments.