Robust Counterparts of Inequalities Containing Sums of Maxima of Linear Functions

This paper adresses the robust counterparts of optimization problems containing sums of maxima of linear functions and proposes several reformulations. These problems include many practical problems, e.g. problems with sums of absolute values, and arise when taking the robust counterpart of a linear inequality that is affine in the decision variables, affine in a parameter with box uncertainty, and affine in a parameter with general uncertainty. In the literature, often the reformulation that is exact when there is no uncertainty is used. However, in robust optimization this reformulation gives an inferior solution and provides a pessimistic view. We observe that in many papers this conservatism is not mentioned. Some papers have recognized this problem, but existing solutions are either too conservative or their performance for different uncertainty regions is not known, a comparison between them is not available, and they are restricted to specific problems. We provide techniques for general problems and compare them with numerical examples in inventory management, regression and brachytherapy. Based on these examples, we give tractable recommendations for reducing the conservatism.robust optimization;sum of maxima of linear functions;biaffine uncertainty;robust conic quadratic constraints

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Inventory management based on target-oriented robust optimization

We propose a target-oriented robust optimization approach to solve a multiproduct, multiperiod inventory management problem subject to ordering capacity constraints. We assume the demand for each product in each period is characterized by an uncertainty set that depends only on a reference value and the bounds of the demand. Our goal is to find an ordering policy that maximizes the sizes of all the uncertainty sets such that all demand realizations from the sets will result in a total cost lower than a prespecified cost target. We prove that a static decision rule is optimal for an approximate formulation of the problem, which significantly reduces the computation burden. By tuning the cost target, the resultant policy can achieve a balance between the expected cost and the associated cost variance. Numerical experiments suggest that although only limited demand information is used, the proposed approach performs comparably to traditional methods based on dynamic programming and stochastic programming. More importantly, our approach significantly outperforms the traditional methods if the latter assume inaccurate demand distributions. We demonstrate the applicability of our approach through two case studies from different industries. This paper was accepted by Yinyu Ye, optimization. </jats:p

Robust Optimization for SCED in AC-HVDC Power Systems

Wind power is a clean, renewable and low-carbon resource for power generation that has received increasing attention in power systems over the last few decades. There are two main challenges associated with the large-scale integration of wind power plants in the power system: i) the intermittent nature of wind power results in prediction errors that can greatly impact the system's operational security and reliability requirements, and ii) large-scale offshore wind farms are typically located far from onshore loads and require new developments in the transmission system of power grids, e.g., realization of mixed alternating current-high voltage direct current (AC-HVDC) power systems, which will introduce new reliability requirements to the system operator. The security-constrained economic dispatch (SCED) problem deals with determining a power dispatch schedule, for all generating units, that minimizes the total operational cost, while taking into account system reliability requirements. Robust optimization (RO) has recently been used to tackle wind power uncertainty in the SCED problem. In the literature of RO, the budget of uncertainty was proposed to adjust the solution conservatism (robustness) such that higher budgets of uncertainty correspond to more conservative solutions. This thesis shows that the budget of uncertainty approach may not be meaningful for problems with RHS uncertainty since increasing the budget of uncertainty by more than a certain threshold may not always impact the level of conservatism. This thesis proposes a new tractable two-stage robust optimization model that effectively incorporates the budget of uncertainty in problems with RHS uncertainty, controls the level of conservatism, and provides meaningful insights on the trade-off between robustness and cost. Furthermore, this thesis examines the applicability of the proposed robust approach for the SCED problem in mixed AC-HVDC power systems with large integration of wind power. The proposed robust SCED model considers the impact of wind power curtailment on the operational cost and reliability requirements of the system. Extensive numerical studies are provided to demonstrate the economic and operational advantages of the proposed robust SCED model in mixed AC-HVDC systems from five aspects: the effectiveness of the budget of uncertainty, robustness against uncertainty, contribution to real-time reliability, cost efficiency, and power transfer controllability

Dynamic Pricing through Sampling Based Optimization

In this paper we develop an approach to dynamic pricing that combines ideas from data-driven and robust optimization to address the uncertain and dynamic aspects of the problem. In our setting, a firm off ers multiple products to be sold over a fixed discrete time horizon. Each product sold consumes one or more resources, possibly sharing the same resources among di fferent products. The firm is given a fixed initial inventory of these resources and cannot replenish this inventory during the selling season. We assume there is uncertainty about the demand seen by the fi rm for each product and seek to determine a robust and dynamic pricing strategy that maximizes revenue over the time horizon. While the traditional robust optimization models are tractable, they give rise to static policies and are often too conservative. The main contribution of this paper is the exploration of closed-loop pricing policies for di fferent robust objectives, such as MaxMin, MinMax Regret and MaxMin Ratio. We introduce a sampling based optimization approach that can solve this problem in a tractable way, with a con fidence level and a robustness level based on the number of samples used. We will show how this methodology can be used for data-driven pricing or adapted for a random sampling optimization approach when limited information is known about the demand uncertainty. Finally, we compare the revenue performance of the di fferent models using numerical simulations, exploring the behavior of each model under diff erent sample sizes and sampling distributions.National Science Foundation (U.S.) (Grant 0556106-CMII)National Science Foundation (U.S.) (Grant 0824674-CMII)Singapore-MIT Allianc