Efficient Kriging-based robust optimization of unconstrained problems

A novel methodology, based on Kriging and expected improvement, is proposed for applying robust optimization on unconstrained problems affected by implementation error. A modified expected improvement measure which reflects the need for robust instead of nominal optimization is used to provide new sampling point locations. A new sample is added at each iteration by finding the location at which the modified expected improvement measure is maximum. By means of this process, the algorithm iteratively progresses towards the robust optimum. It is demonstrated that the algorithm performs significantly better than current techniques for robust optimization using response surface modeling.     

A robust algorithm for quadratic optimization under quadratic constraints

Most existing methods of quadratically constrained quadratic optimization actually solve a refined linear or convex relaxation of the original problem. It turned out, however, that such an approach may sometimes provide an infeasible solution which cannot be accepted as an approximate optimal solution in any reasonable sense. To overcome these limitations a new approach is proposed that guarantees a more appropriate approximate optimal solution which is also stable under small perturbations of the constraints.     

An approximation technique for robust nonlinear optimization

Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that is, however, difficult to treat computationally. In this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set. In case of any norm bounded parameter uncertainty, this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints. The main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show how to generalize the techniques to optimal control problems, and discuss how even infinite dimensional uncertainties can be treated efficiently. Finally, we present optimization results for an example from process engineering, a batch distillation.     

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.     

Distributionally robust discrete optimization with Entropic Value-at-Risk

We study the discrete optimization problem under the distributionally robust framework. We optimize the Entropic Value-at-Risk, which is a coherent risk measure and is also known as Bernstein approximation for the chance constraint. We propose an efficient approximation algorithm to resolve the problem via solving a sequence of nominal problems. The computational results show that the number of nominal problems required to be solved is small under various distributional information sets.     

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.      

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.     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

Algorithm robust for the bicriteria discrete optimization problem

We apply Algorithm Robust to various problems in multiple objective discrete optimization. Algorithm Robust is a general procedure that is designed to solve bicriteria optimization problems. The algorithm performs a weight space search in which the weights are utilized in min-max type subproblems. In this paper, we experiment with Algorithm Robust on the bicriteria knapsack problem, the bicriteria assignment problem, and the bicriteria minimum cost network flow problem. We look at a heuristic variation that is based on controlling the weight space search and has an indirect control on the sample of efficient solutions generated. We then study another heuristic variation which generates samples of the efficient set with quality guarantees. We report results of computational experiments.     

Approximating infinite horizon stochastic optimal control in discrete time with constraints

Traditional approaches to solving stochastic optimal control problems involve dynamic programming, and solving certain optimality equations. When recast as stochastic programming problems, structural aspects such as convexity are retained, and numerical solution procedures based on decomposition and duality may be exploited. This paper explores a class of stationary, infinite-horizon stochastic optimization problems with discounted cost criterion. Constraints on both states and controls are permitted, and modeled in the objective function by allowing it to take infinite values. Approximating techniques are developed using variational analysis, and intuitive lower bounds are obtained via averaging the future. These bounds could be used in a finite-time horizon stochastic programming setting to find solutions numerically. Research supported in part by a grant of the National Science Foundation. AMS Classification 46N10, 49N15, 65K10, 90C15, 90C46     

Boundaries of the receding horizon control for interconnected systems

In recent years, the finite-horizon quadratic minimization problem has become popular in process control, where the horizon is constantly rolled back. In this paper, this type of control, which is also called the receding horizon control, is considered for interconnected systems. First, the receding horizon control equations are formulated; then, some stability conditions depending on the interconnection norms and the horizon lengths are presented. For -coupled systems, stability results similar to centralized systems are obtained. For interconnected systems which are not -coupled, the existence of a horizon length and a corresponding stabilizing receding horizon control are derived. Finally, the performance of a locally computed receding horizon control for time-invariant and time-varying systems with different updating intervals is examined in an example.     

A robust approach for modeling limited observability in bilevel optimization

Many applications of bilevel optimization contain a leader facing a follower whose reaction deviates from the one expected by the leader due to some kind of bounded rationality. We consider bilinear bilevel problems with follower's response uncertainty due to limited observability regarding the leader's decision and exploit robust optimization to model the decision making of the follower. We show that the robust counterpart of the lower level allows to tackle the problem via the lower level's KKT conditions.     

Second-order sufficient conditions for control problems with mixed control-state constraints

References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.The authors wish to thank K. Malanowski for helpful discussions.     

Proportion-based robust optimization and team orienteering problem with interval data

In this paper, a proportion-based robust optimization approach is developed to deal with uncertain combinatorial optimization problems. This approach assumes that a certain proportion of uncertain coefficients in each solution are allowed to change and optimizes a deterministic model so as to achieve a trade-off between optimality and feasibility when the coefficients change. We apply this approach on team orienteering problem with interval data (TOPID), a variant of vehicle routing problem, which has not yet been studied before. A branch and price algorithm is proposed to solve the robust counterpart by using two novel dominance relations. Finally, numerical study is performed. The results show the usefulness of the proposed robust optimization approach and the effectiveness of our algorithm.     

Conditions implying the vanishing of the Hamiltonian at infinity in optimal control problems

In an infinite horizon optimal control problem the Hamiltonian vanishes at infinity when the differential equation is autonomous and the integrand in the criterion satisfies some weak integrability conditions. A generalization of Michel’s result (in Econometrica 50:975–985, 1982) is obtained.     

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.     

Augmented Lagrangian method for distributed optimal control problems with state constraints

We consider state-constrained optimal control problems governed by elliptic equations. Doing Slater-like assumptions, we know that Lagrange multipliers exist for such problems, and we propose a decoupled augmented Lagrangian method. We present the algorithm with a simple example of a distributed control problem.     

Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints

This paper studies multiobjective optimal control problems in presence of constraints in the discrete time framework. Both the finite- and infinite-horizon settings are considered. The paper provides necessary conditions of Pareto optimality under lighter smoothness assumptions compared to the previously obtained results. These conditions are given in the form of weak and strong Pontryagin principles which generalize the existing ones. To obtain some of these results, we provide new multiplier rules for multiobjective static optimization problems and new Pontryagin principles for the finite horizon multiobjective optimal control problems.     

A robust optimization model for a cross-border logistics problem with fleet composition in an uncertain environment

Since the implementation of the open-door policy in China, many Hong Kong-based manufacturers' production lines have moved to China to take advantage of the lower production cost, lower wages, and lower rental costs, and thus, the finished products must be transported from China to Hong Kong. It has been discovered that logistics management often encounters uncertainty and noisy data. In this paper, a robust optimization model is proposed to solve a cross-border logistics problem in an environment of uncertainty. By adjusting penalty parameters, decision-makers can determine an optimal long-term transportation strategy, including the optimal delivery routes and the optimal vehicle fleet composition to minimize total expenditure under different economic growth scenarios. We demonstrate the robustness and effectiveness of our model using the example of a Hong Kong-based manufacturing company. The analysis of the trade-off between model robustness and solution robustness is also presented.