Bram L. Gorissen

A practical guide to robust optimization

Robust optimization is a young and active research field that has been mainly developed in the last 15 years. Robust optimization is very useful for practice, since it is tailored to the information at hand, and it leads to computationally tractable formulations. It is therefore remarkable that real-life applications of robust optimization are still lagging behind; there is much more potential for real-life applications than has been exploited hitherto. The aim of this paper is to help practitioners to understand robust optimization and to successfully apply it in practice. We provide a brief introduction to robust optimization, and also describe important do׳s and don׳ts for using it in practice. We use many small examples to illustrate our discussions.

Technical Note—Deriving Robust and Globalized Robust Solutions of Uncertain Linear Programs with General Convex Uncertainty Sets

We propose a new way to derive tractable robust counterparts of a linear program based on the duality between the robust (“pessimistic”) primal problem and its “optimistic” dual. First we obtain a new convex reformulation of the dual problem of a robust linear program, and then show how to construct the primal robust solution from the dual optimal solution. Our result allows many new uncertainty regions to be considered. We give examples of tractable uncertainty regions that were previously intractable. The results are illustrated by solving a multi-item newsvendor problem. We also apply the new method to the globalized robust counterpart scheme and show its tractability.

High-dose-rate prostate brachytherapy inverse planning on dose-volume criteria by simulated annealing.

High-dose-rate brachytherapy is a tumor treatment method where a highly radioactive source is brought in close proximity to the tumor. In this paper we develop a simulated annealing algorithm to optimize the dwell times at preselected dwell positions to maximize tumor coverage under dose-volume constraints on the organs at risk. Compared to existing algorithms, our algorithm has advantages in terms of speed and objective value and does not require an expensive general purpose solver. Its success mainly depends on exploiting the efficiency of matrix multiplication and a careful selection of the neighboring states. In this paper we outline its details and make an in-depth comparison with existing methods using real patient data.

Dwell time modulation restrictions do not necessarily improve treatment plan quality for prostate HDR brachytherapy

Inverse planning algorithms for dwell time optimisation in interstitial high-dose-rate (HDR) brachytherapy may produce solutions with large dwell time variations within catheters, which may result in undesirable selective high-dose subvolumes. Extending the dwell time optimisation model with a dwell time modulation restriction (DTMR) that limits dwell time differences between neighboring dwell positions has been suggested to eliminate this problem. DTMRs may additionally reduce the sensitivity for uncertainties in dwell positions that inevitably result from catheter reconstruction errors and afterloader source positioning inaccuracies. This study quantifies the reduction of high-dose subvolumes and the robustness against these uncertainties by applying a DTMR to template-based prostate HDR brachytherapy implants. Three different DTMRs were consecutively applied to a linear dose-based penalty model (LD) and a dose-volume based model (LDV), both obtained from literature. The models were solved with DTMR levels ranging from no restriction to uniform dwell times within catheters in discrete steps. Uncertainties were simulated on clinical cases using in-house developed software, and dose-volume metrics were calculated in each simulation. For the assessment of high-dose subvolumes, the dose homogeneity index (DHI) and the contiguous dose volume histogram were analysed. Robustness was measured by the improvement of the lowest D90% of the planning target volume (PTV) observed in the simulations. For (LD), a DTMR yields an increase in DHI of approximately 30% and reduces the size of the largest high-dose volume by 2-5 cc. However, this comes at a cost of a reduction in D90% of the PTV of 10%, which often implies that it drops below the desired minimum of 100%. For (LDV), none of the DTMRs were able to improve high-dose volume measures. DTMRs were not capable of improving robustness of PTV D90% against uncertainty in dwell positions for both models.Comparison of optimization algorithms for inverse treatment planning requires objective function value evaluation.

Robust Fractional Programming

We extend robust optimization (RO) to fractional programming, where both the objective and the constraints contain uncertain parameters. Earlier work did not consider uncertainty in both the objective and the constraints, or did not use RO. Our contribution is threefold. First, we provide conditions to guarantee that either a globally optimal solution, or a sequence converging to the globally optimal solution, can be found by solving one or more convex optimization problems. Second, we identify two cases for which an exact solution can be obtained by solving a single optimization problem: (1) when uncertainty in the numerator is independent from the uncertainty in the denominator, and (2) when the denominator does not contain an optimization variable. Third, we show that the general problem can be solved with an (iterative) root finding method. The results are demonstrated on a return on investment maximization problem, data envelopment analysis, and mean-variance optimization. We find that the robust optimal solution is only slightly more robust than the nominal solution. As a side-result, we use RO to show that two existing methods for solving fractional programs are dual to each other.

A New Method for Deriving Robust and Globalized Robust Solutions of Uncertain Linear Conic Optimization Problems Having General Convex Uncertainty Sets

Abstract: We propose a new way to derive tractable robust counterparts of a linear conic optimization problem by using the theory of Beck and Ben-Tal [2] on the duality between the robust (“pessimistic”) primal problem and its “optimistic” dual. First, we obtain a new convex reformulation of the dual problem of a robust linear conic program, and then show how to construct the primal robust solution from the dual optimal solution. Our result allows many new uncertainty regions to be considered. We give examples of tractable uncertainty regions that were previously intractable. The results are illustrated by solving a multi-item newsvendor problem. We also propose a new globalized robust counterpart that is more flexible, and is tractable for general convex uncertainty sets and any convex distance function.

Comment on ‘Comparative evaluation of two dose optimization methods for image-guided, highly-conformal, tandem and ovoids cervix brachytherapy planning’

With great interest we read the recently published article by Ren et al. (2013) on anatomy-based inverse dose optimization for tandem and ovoids cervix brachytherapy planning. The article compares Nelder-Mead Simplex (NMS) with Simulated Annealing (SA) for dwell time optimization, and shows that SA is superior because of better organ-at-risk sparing, lower dwell time variability and smaller sensitivity on the starting point of initial dwell times.