Generalized light robustness and the trade-off between robustness and nominal quality

Robust optimization considers optimization problems with uncertainty in the data. The common data model assumes that the uncertainty can be represented by an uncertainty set. Classic robust optimization considers the solution under the worst case scenario. The resulting solutions are often too conservative, e.g. they have high costs compared to non-robust solutions. This is a reason for the development of less conservative robust models. In this paper we extract the basic idea of the concept of light robustness originally developed in Fischetti and Monaci (Robust and online large-scale optimization, volume 5868 of lecture note on computer science. Springer, Berlin, pp 61–84, 2009) for interval-based uncertainty sets and linear programs: fix a quality standard for the nominal solution and among all solutions satisfying this standard choose the most reliable one. We then use this idea in order to formulate the concept of light robustness for arbitrary optimization problems and arbitrary uncertainty sets. We call the resulting concept generalized light robustness. We analyze the concept and discuss its relation to other well-known robustness concepts such as strict robustness (Ben-Tal et al. in Robust optimization. Princeton University Press, Princeton, 2009), reliability (Ben-Tal and Nemirovski in Math Program A 88:411–424, 2000) or the approach of Bertsimas and Sim (Oper Res 52(1):35–53, 2004). We show that the light robust counterpart is computationally tractable for many different types of uncertainty sets, among them polyhedral or ellipsoidal uncertainty sets. We furthermore discuss the trade-off between robustness and nominal quality and show that non-dominated solutions with respect to nominal quality and robustness can be computed by the generalized light robustness approach.     

Cutting plane versus compact formulations for uncertain (integer) linear programs

Robustness is about reducing the feasible set of a given nominal optimization problem by cutting ??risky?? solutions away. To this end, the most popular approach in the literature is to extend the nominal model with a polynomial number of additional variables and constraints, so as to obtain its robust counterpart. Robustness can also be enforced by adding a possibly exponential family of cutting planes, which typically leads to an exponential formulation where cuts have to be generated at run time. Both approaches have pros and cons, and it is not clear which is the best one when approaching a specific problem. In this paper we computationally compare the two options on some prototype problems with different characteristics. We first address robust optimization à la Bertsimas and Sim for linear programs, and show through computational experiments that a considerable speedup (up to 2 orders of magnitude) can be achieved by exploiting a dynamic cut generation scheme. For integer linear problems, instead, the compact formulation exhibits a typically better performance. We then move to a probabilistic setting and introduce the uncertain set covering problem where each column has a certain probability of disappearing, and each row has to be covered with high probability. A related uncertain graph connectivity problem is also investigated, where edges have a certain probability of failure. For both problems, compact ILP models and cutting plane solution schemes are presented and compared through extensive computational tests. The outcome is that a compact ILP formulation (if available) can be preferable because it allows for a better use of the rich arsenal of preprocessing/cut generation tools available in modern ILP solvers. For the cases where such a compact ILP formulation is not available, as in the uncertain connectivity problem, we propose a restart solution strategy and computationally show its practical effectiveness.     

A note on the Bertsimas & Sim algorithm for robust combinatorial optimization problems

We improve the well-known result presented in Bertsimas and Sim (Math Program B98:49–71, 2003) regarding the computation of optimal solutions of Robust Combinatorial Optimization problems with interval uncertainty in the objective function coefficients. We also extend this improvement to a more general class of Combinatorial Optimization problems with interval uncertainty.     

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.     

Robust combinatorial optimization with variable budgeted uncertainty

We introduce a new model for robust combinatorial optimization where the uncertain parameters belong to the image of multifunctions of the problem variables. In particular, we study the variable budgeted uncertainty, an extension of the budgeted uncertainty introduced by Bertsimas and Sim. Variable budgeted uncertainty can provide the same probabilistic guarantee as the budgeted uncertainty while being less conservative for vectors with few non-zero components. The feasibility set of the resulting optimization problem is in general non-convex so that we propose a mixed-integer programming reformulation for the problem, based on the dualization technique often used in robust linear programming. We show how to extend these results to non-binary variables and to more general multifunctions involving uncertainty set defined by conic constraints that are affine in the problem variables. We present a computational comparison of the budgeted uncertainty and the variable budgeted uncertainty on the robust knapsack problem. The experiments show a reduction of the price of robustness by an average factor of 18 %.     

A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems

In this paper we present a genetic algorithm-based heuristic especially for the weighted maximum independent set problem (IS). The proposed approach treats also some equivalent combinatorial optimization problems. We introduce several modifications to the basic genetic algorithm, by (i) using a crossover called two-fusion operator which creates two new different children and (ii) replacing the mutation operator by the heuristic-feasibility operator tailored specifically for the weighted independent set. The performance of our algorithm was evaluated on several randomly generated problem instances for the weighted independent set and on some instances of the DIMACS Workshop for the particular case: the unweighted maximum clique problem. Computational results show that the proposed approach is able to produce high-quality solutions within reasonable computational times. This algorithm is easily parallelizable and this is one of its important features.     

Single machine earliness and tardiness scheduling

We examine the problem of scheduling a given set of jobs on a single machine to minimize total early and tardy costs without considering machine idle time. We decompose the problem into two subproblems with a simpler structure. Then the lower bound of the problem is the sum of the lower bounds of two subproblems. A lower bound of each subproblem is obtained by Lagrangian relaxation. Rather than using the well-known subgradient optimization approach, we develop two efficient multiplier adjustment procedures with complexity O(nlog n) to solve two Lagrangian dual subproblems. A branch-and-bound algorithm based on the two efficient procedures is presented, and is used to solve problems with up to 50 jobs, hence doubling the size of problems that can be solved by existing branch-and-bound algorithms. We also propose a heuristic procedure based on the neighborhood search approach. The computational results for problems with up to 3 000 jobs show that the heuristic procedure performs much better than known heuristics for this problem in terms of both solution efficiency and quality. In addition, the results establish the effectiveness of the heuristic procedure in solving realistic problems to optimality or near optimality.     

Lagrangian relaxation guided problem space search heuristics for generalized assignment problems

We develop and test a heuristic based on Lagrangian relaxation and problem space search to solve the generalized assignment problem (GAP). The heuristic combines the iterative search capability of subgradient optimization used to solve the Lagrangian relaxation of the GAP formulation and the perturbation scheme of problem space search to obtain high-quality solutions to the GAP. We test the heuristic using different upper bound generation routines developed within the overall mechanism. Using the existing problem data sets of various levels of difficulty and sizes, including the challenging largest instances, we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high-quality solutions quickly. An advantage of the approach is its generic nature, simplicity, and implementation flexibility.     

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.      

The balanced minimum evolution problem under uncertain data

We investigate the Robust Deviation Balanced Minimum Evolution Problem (RDBMEP), a combinatorial optimization problem that arises in computational biology when the evolutionary distances from taxa are uncertain and varying inside intervals. By exploiting some fundamental properties of the objective function, we present a mixed integer programming model to exactly solve instances of the RDBMEP and discuss the biological impact of uncertainty on the solutions to the problem. Our results give perspective on the mathematics of the RDBMEP and suggest new directions to tackle phylogeny estimation problems affected by uncertainty.     

The equitable location problem on the plane

This paper considers the problem of locating M facilities on the unit square so as to minimize the maximal demand faced by each facility subject to closest assignments and coverage constraints. Focusing on uniform demand over the unit square, we develop upper and lower bounds on feasibility of the problem for a given number of facilities and coverage radius. Based on these bounds and numerical experiments we suggest a heuristic to solve the problem. Our computational results show that the heuristic is very efficient, as the average gap between its solutions and the lower bound is 4.34%.     

A discussion on the conservatism of robust linear optimization problems

In 2004, Bertsimas and Sim proposed a robust approach that can control the degree of conservatism by applying a limitation Γ to the maximum number of parameters that are allowed to change. However, the robust approach can become extremely conservative even when Γ is relatively small. In this paper, we provide a theoretical analysis to explain why this extreme conservatism occurs. We further point out that the robust approach does not reach an extremely conservative state when Γ is less than k, where k is the number of nonzero components of the optimal solution of the extremely conservative robust approach. This research also shows that care must be taken when adjusting the value of Γ to control the degree of conservatism because the approach may result in greater conservatism than was intended. We subsequently apply our analysis to additive combinatorial optimization problems. Finally, we illustrate our results on numerical simulations.     

Robust combinatorial optimization with variable cost uncertainty

We present in this paper a new model for robust combinatorial optimization with cost uncertainty that generalizes the classical budgeted uncertainty set. We suppose here that the budget of uncertainty is given by a function of the problem variables, yielding an uncertainty multifunction. The new model is less conservative than the classical model and approximates better Value-at-Risk objective functions, especially for vectors with few non-zero components. An example of budget function is constructed from the probabilistic bounds computed by Bertsimas and Sim. We provide an asymptotically tight bound for the cost reduction obtained with the new model. We turn then to the tractability of the resulting optimization problems. We show that when the budget function is affine, the resulting optimization problems can be solved by solving n+1$n+1$ deterministic problems. We propose combinatorial algorithms to handle problems with more general budget functions. We also adapt existing dynamic programming algorithms to solve faster the robust counterparts of optimization problems, which can be applied both to the traditional budgeted uncertainty model and to our new model. We evaluate numerically the reduction in the price of robustness obtained with the new model on the shortest path problem and on a survivable network design problem.     

考虑共享不确定因素的应急设施最大覆盖选址模型

为提升应急设施的服务质量和抵御中断风险的能力,研究应急设施最大覆盖选址-分配决策问题。扩展无容量限制的固定费用的可靠性选址决策模型,建立考虑共享不确定因素的应急设施最大覆盖选址优化模型,通过在目标和约束中引入budget不确定集刻画共享不确定因素,基于Bertsimas和Sim鲁棒优化方法建立混合整数规划模型,并将非线性问题转化为易于求解的鲁棒等价模型,利用带混沌搜索策略的改进灰狼优化算法求解模型,并对不确定鲁棒水平和中断概率进行敏感性分析。最后通过案例及数据仿真结果的对比分析,验证了模型的合理性和有效性,并给出最优的选址分配布局。     

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.     

A novel differential evolution algorithm for binary optimization

Differential evolution (DE) is one of the most powerful stochastic search methods which was introduced originally for continuous optimization. In this sense, it is of low efficiency in dealing with discrete problems. In this paper we try to cover this deficiency through introducing a new version of DE algorithm, particularly designed for binary optimization. It is well-known that in its original form, DE maintains a differential mutation, a crossover and a selection operator for optimizing non-linear continuous functions. Therefore, developing the new binary version of DE algorithm, calls for introducing operators having the major characteristics of the original ones and being respondent to the structure of binary optimization problems. Using a measure of dissimilarity between binary vectors, we propose a differential mutation operator that works in continuous space while its consequence is used in the construction of the complete solution in binary space. This approach essentially enables us to utilize the structural knowledge of the problem through heuristic procedures, during the construction of the new solution. To verify effectiveness of our approach, we choose the uncapacitated facility location problem (UFLP)—one of the most frequently encountered binary optimization problems—and solve benchmark suites collected from OR-Library. Extensive computational experiments are carried out to find out the behavior of our algorithm under various setting of the control parameters and also to measure how well it competes with other state of the art binary optimization algorithms. Beside UFLP, we also investigate the suitably of our approach for optimizing numerical functions. We select a number of well-known functions on which we compare the performance of our approach with different binary optimization algorithms. Results testify that our approach is very efficient and can be regarded as a promising method for solving wide class of binary optimization problems.     

Using Lagrangian dual information to generate degree constrained spanning trees

In this paper, a Lagrangian-based heuristic is proposed for the degree constrained minimum spanning tree problem. The heuristic uses Lagrangian relaxation information to guide the construction of feasible solutions to the problem. The scheme operates, within a Lagrangian relaxation framework, with calls to a greedy construction heuristic, followed by a heuristic improvement procedure. A look ahead infeasibility prevention mechanism, introduced into the greedy heuristic, allowed us to solve instances of the problem where some of the vertices are restricted to having degrees 1 or 2. Furthermore, in order to cut down on CPU time, a restricted version of the original problem is formulated and used to generate feasible solutions. Extensive computational experiments were conducted and indicate that the proposed heuristic is competitive with the best heuristics and metaheuristics in the literature.     

Robust portfolio optimization with a hybrid heuristic algorithm

Estimation errors in both the expected returns and the covariance matrix hamper the construction of reliable portfolios within the Markowitz framework. Robust techniques that incorporate the uncertainty about the unknown parameters are suggested in the literature. We propose a modification as well as an extension of such a technique and compare both with another robust approach. In order to eliminate oversimplifications of Markowitz’ portfolio theory, we generalize the optimization framework to better emulate a more realistic investment environment. Because the adjusted optimization problem is no longer solvable with standard algorithms, we employ a hybrid heuristic to tackle this problem. Our empirical analysis is conducted with a moving time window for returns of the German stock index DAX100. The results of all three robust approaches yield more stable portfolio compositions than those of the original Markowitz framework. Moreover, the out-of-sample risk of the robust approaches is lower and less volatile while their returns are not necessarily smaller.     

Combined vehicle routing and scheduling with temporal precedence and synchronization constraints

We present a mathematical programming model for the combined vehicle routing and scheduling problem with time windows and additional temporal constraints. The temporal constraints allow for imposing pairwise synchronization and pairwise temporal precedence between customer visits, independently of the vehicles. We describe some real world problems where in the literature the temporal constraints are usually remarkably simplified in the solution process, even though these constraints may significantly improve the solution quality and/or usefulness. We also propose an optimization based heuristic to solve real size instances. The results of numerical experiments substantiate the importance of the temporal constraints in the solution approach. We also make a computational study by comparing a direct use of a commercial solver against the proposed heuristic, where the latter approach can find high quality solutions within specific time limits.     

Minimizing makespan with multiple-orders-per-job in a two-machine flowshop

We investigate a new scheduling problem, multiple-orders-per-job (MOJ), in the context of a two-machine flowshop. Lower bounds for the makespan performance measure are provided for combinations of lot-processing and item-processing machines. An optimization model is presented that addresses both job formation and job sequencing. We define a heuristic to minimize the makespan for the MOJ problem for two-machine item-processing flowshops. The heuristic obtains solutions within 2% of a tight lower bound and runs in O(HF) time, where H is the number of orders and F is the restricted number of jobs.