A robust robust optimization result

We study the loss in objective value when an inaccurate objective is optimized instead of the true one, and show that “on average” this loss is very small, for an arbitrary compact feasible region.     

The dynamic programming approach to multi-model robust optimization

The aim of this paper is to extend the dynamic programming (DP) approach to multi-model optimal control problems (OCPs). We deal with robust optimization of multi-model control systems and are particularly interested in the Hamilton-Jacobi-Bellman (HJB) equation for the above class of problems. In this paper, we study a variant of the HJB for multi-model OCPs and examine the natural relationship between the Bellman DP techniques and the Robust Maximum Principle (MP). Moreover, we describe how to carry out the practical calculations in the context of multi-model LQ-problems and derive the associated Riccati-type equation.     

A dynamic programming approach to adjustable robust optimization

In this paper we consider the adjustable robust approach to multistage optimization, for which we derive dynamic programming equations. We also discuss this from the point of view of risk averse stochastic programming. We consider as an example a robust formulation of the classical inventory model and show that, like for the risk neutral case, a basestock policy is optimal.     

Efficient robust optimization for robust control with constraints

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.     

Optimization for automated assembly of puzzles

The puzzle-assembly problem has many application areas such as restoration and reconstruction of archeological findings, repairing of broken objects, solving jigsaw type puzzles, molecular docking problem, etc. The puzzle pieces usually include not only geometrical shape information but also visual information such as texture, color, and continuity of lines. This paper presents a new approach to the puzzle-assembly problem that is based on using textural features and geometrical constraints. The texture of a band outside the border of pieces is predicted by inpainting and texture synthesis methods. Feature values are derived from these original and predicted images of pieces. An affinity measure of corresponding pieces is defined and alignment of the puzzle pieces is formulated as an optimization problem where the optimum assembly of the pieces is achieved by maximizing the total affinity measure. A Fast Fourier Transform based image registration technique is used to speed up the alignment of the pieces. Experimental results are presented on real and artificial data sets.     

A robust optimization approach to wine grape harvesting scheduling

Optimization models are increasingly being used in agricultural planning. However, the inherent uncertainties present in agriculture make it difficult. In recent years, robust optimization has emerged as a methodology that allows dealing with uncertainty in optimization models, even when probabilistic knowledge of the phenomenon is incomplete. In this paper, we consider a wine grape harvesting scheduling optimization problem subject to several uncertainties, such as the actual productivity that can be achieved when harvesting. We study how effective robust optimization is solving this problem in practice. We develop alternative robust models and show results for some test problems obtained from actual wine industry problems.     

A decision-theoretic approach to robust optimization in multivalued graphs

This paper is devoted to the search of robust solutions in finite graphs when costs depend on scenarios. We first point out similarities between robust optimization and multiobjective optimization. Then, we present axiomatic requirements for preference compatibility with the intuitive idea of robustness in a multiple scenarios decision context. This leads us to propose the Lorenz dominance rule as a basis for robustness analysis. Then, after presenting complexity results about the determination of Lorenz optima, we show how the search can be embedded in algorithms designed to enumerate k best solutions. Then, we apply it in order to enumerate Lorenz optimal spanning trees and paths. We investigate possible refinements of Lorenz dominance and we propose an axiomatic justification of OWA operators in this context. Finally, the results of numerical experiments on randomly generated graphs are provided. They show the numerical efficiency of the suggested approach.     

Surrogate duality for robust optimization

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min–max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, and show some examples at which such duality results are used effectively. Moreover, we obtain a surrogate duality theorem and a surrogate min–max duality theorem for semi-definite optimization problems in the face of data uncertainty.     

Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem

$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$

where f is a polynomial and \({\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}\) as well as \({{\Omega}\subset\mathbb{R}^p}\) are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations \({(J_i)\subset\mathbb{R}[\mathbf{y}]}\) of the optimal value function \({J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}\). The polynomial \({J_i\in\mathbb{R}[\mathbf{y}]}\) is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem \({\mathbf{y}\mapsto J(\mathbf{y})}\), recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem \({J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}\) and prove that \({\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}\) converges to J* as i → ∞. Finally, for fixed ? ≤ i, each \({J^*_\ell}\) (and hence \({\hat{J}^*_i}\)) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).

     

On robust optimization of two-stage systems

Robust-optimization models belong to a special class of stochastic programs, where the traditional expected cost minimization objective is replaced by one that explicitly addresses cost variability. This paper explores robust optimization in the context of two-stage planning systems. We show that, under arbitrary measures for variability, the robust optimization approach might lead to suboptimal solutions to the second-stage planning problem. As a result, the variability of the second-stage costs may be underestimated, thereby defeating the intended purpose of the model. We propose sufficient conditions on the variability measure to remedy this problem. Under the proposed conditions, a robust optimization model can be efficiently solved using a variant of the L-shaped decomposition algorithm for traditional stochastic linear programs. We apply the proposed framework to standard stochastic-programming test problems and to an application that arises in auctioning excess electric power. Mathematics Subject Classification (1991):90C15, 90C33, 90B50, 90A09, 90A43Supported in part by NSF Grants DMI-0099726 and DMI-0133943     

Decomposition for adjustable robust linear optimization subject to uncertainty polytope

We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max–min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.     

Adjustable robust optimization through multi-parametric programming

Adjustable robust optimization (ARO) involves recourse decisions (i.e. reactive actions after the realization of the uncertainty, ‘wait-and-see’     

Likelihood robust optimization for data-driven problems

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.     

On distributionally robust multiperiod stochastic optimization

This paper considers model uncertainty for multistage stochastic programs. The data and information structure of the baseline model is a tree, on which the decision problem is defined. We consider “ambiguity neighborhoods” around this tree as alternative models which are close to the baseline model. Closeness is defined in terms of a distance for probability trees, called the nested distance. This distance is appropriate for scenario models of multistage stochastic optimization problems as was demonstrated in Pflug and Pichler (SIAM J Optim 22:1–23, 2012). The ambiguity model is formulated as a minimax problem, where the the optimal decision is to be found, which minimizes the maximal objective function within the ambiguity set. We give a setup for studying saddle point properties of the minimax problem. Moreover, we present solution algorithms for finding the minimax decisions at least asymptotically. As an example, we consider a multiperiod stochastic production/inventory control problem with weekly ordering. The stochastic scenario process is given by the random demands for two products. We determine the minimax solution and identify the worst trees within the ambiguity set. It turns out that the probability weights of the worst case trees are concentrated on few very bad scenarios.     

Quadratic optimization problems in robust beamforming

This paper presents a class of constrained optimization problems whereby a quadratic cost function is to be minimized with respect to a weight vector subject to an inequality quadratic constraint on the weight vector. This class of constrained optimization problems arises as a result of a motivation for designing robust antenna array processors in the field of adaptive array processing. The constrained optimization problem is first solved by using the primal-dual method. Numerical techniques are presented to reduce the computational complexity of determining the optimal Lagrange multiplier and hence the optimal weight vector. Subsequently, a set of linear constraints or at most linear plus norm constraints are developed for approximating the performance achievable with the quadratic constraint. The use of linear constraints is very attractive, since they reduce the computational burden required to determine the optimal weight vector.     

Semiconductor lot allocation using robust optimization

In this work, the problem of allocating a set of production lots to satisfy customer orders is considered. This research is of relevance to lot-to-order matching problems in semiconductor supply chain settings. We consider that lot-splitting is not allowed during the allocation process due to standard practices. Furthermore, lot-sizes are regarded as uncertain planning data when making the allocation decisions due to potential yield loss. In order to minimize the total penalties of demand un-fulfillment and over-fulfillment, a robust mixed-integer optimization approach is adopted to model is proposed the problem of allocating a set of work-in-process lots to customer orders, where lot-sizes are modeled using ellipsoidal uncertainty sets. To solve the optimization problem efficiently we apply the techniques of branch-and-price and Benders decomposition. The advantages of our model are that it can represent uncertainty in a straightforward manner with little distributional assumptions, and it can produce solutions that effectively hedge against the uncertainty in the lot-sizes using very reasonable amounts of computational effort.     

Selected topics in robust convex optimization

Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-but- bounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control.      

Reformulation versus cutting-planes for robust optimization

Robust optimization (RO) is a tractable method to address uncertainty in optimization problems where uncertain parameters are modeled as belonging to uncertainty sets that are commonly polyhedral or ellipsoidal. The two most frequently described methods in the literature for solving RO problems are reformulation to a deterministic optimization problem or an iterative cutting-plane method. There has been limited comparison of the two methods in the literature, and there is no guidance for when one method should be selected over the other. In this paper we perform a comprehensive computational study on a variety of problem instances for both robust linear optimization (RLO) and robust mixed-integer optimization (RMIO) problems using both methods and both polyhedral and ellipsoidal uncertainty sets. We consider multiple variants of the methods and characterize the various implementation decisions that must be made. We measure performance with multiple metrics and use statistical techniques to quantify certainty in the results. We find for polyhedral uncertainty sets that neither method dominates the other, in contrast to previous results in the literature. For ellipsoidal uncertainty sets we find that the reformulation is better for RLO problems, but there is no dominant method for RMIO problems. Given that there is no clearly dominant method, we describe a hybrid method that solves, in parallel, an instance with both the reformulation method and the cutting-plane method. We find that this hybrid approach can reduce runtimes to 50–75 % of the runtime for any one method and suggest ways that this result can be achieved and further improved on.