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.     

Robust mean variance optimization problem under Rényi divergence information

In this paper, we consider the robust mean variance optimization problem where the probability distribution of assets’ returns is multivariate normal and the uncertain mean and covariance are controlled by a constraint involving Rényi divergence. We present the closed-form solutions for the robust mean variance optimization problem and find that the choice of order parameter which is related to the Rényi divergence measure will not impact optimal portfolio strategy under the cases that the mean vector and the covariance matrix are uncertain, respectively. Moreover, we obtain the closed-form solution for the robust mean variance optimization problem under the case that the mean vector and the covariance matrix are both uncertain. We illustrate the efficiency of our results with an example.     

Support vector machine classifiers with uncertain knowledge sets via robust optimization

In this article we study support vector machine (SVM) classifiers in the face of uncertain knowledge sets and show how data uncertainty in knowledge sets can be treated in SVM classification by employing robust optimization. We present knowledge-based SVM classifiers with uncertain knowledge sets using convex quadratic optimization duality. We show that the knowledge-based SVM, where prior knowledge is in the form of uncertain linear constraints, results in an uncertain convex optimization problem with a set containment constraint. Using a new extension of Farkas' lemma, we reformulate the robust counterpart of the uncertain convex optimization problem in the case of interval uncertainty as a convex quadratic optimization problem. We then reformulate the resulting convex optimization problems as a simple quadratic optimization problem with non-negativity constraints using the Lagrange duality. We obtain the solution of the converted problem by a fixed point iterative algorithm and establish the convergence of the algorithm. We finally present some preliminary results of our computational experiments of the method.     

A robust optimization model for multi-product two-stage capacitated production planning under uncertainty

Production planning (PP) is one of the most important issues carried out in manufacturing environments which seeks efficient planning, scheduling and coordination of all production activities that optimizes the company’s objectives. In this paper, we studied a two-stage real world capacitated production system with lead time and setup decisions in which some parameters such as production costs and customer demand are uncertain. A robust optimization model is developed to formulate the problem in which minimization of the total costs including the setup costs, production costs, labor costs, inventory costs, and workforce changing costs is considered as performance measure. The robust approach is used to reduce the effects of fluctuations of the uncertain parameters with regards to all the possible future scenarios. A mixed-integer programming (MIP) model is developed to formulate the related robust production planning problem. In fact the robust proposed model is presented to generate an initial robust schedule. The performance of this schedule could be improved against of any possible occurrences of uncertain parameters. A case from an Iran refrigerator factory is studied and the characteristics of factory and its products are discussed. The computational results display the robustness and effectiveness of the model and highlight the importance of using robust optimization approach in generating more robust production plans in the uncertain environments. The tradeoff between solution robustness and model robustness is also analyzed.     

Belief linear programming

This paper proposes solution approaches to the belief linear programming (BLP). The BLP problem is an uncertain linear program where uncertainty is expressed by belief functions. The theory of belief function provides an uncertainty measure that takes into account the ignorance about the occurrence of single states of nature. This is the case of many decision situations as in medical diagnosis, mechanical design optimization and investigation problems. We extend stochastic programming approaches, namely the chance constrained approach and the recourse approach to obtain a certainty equivalent program. A generic solution strategy for the resulting certainty equivalent is presented.     

基于鲁棒二阶随机占优的投资组合优化模型研究

本文主要考虑一类经典的含有二阶随机占优约束的投资组合优化问题,其目标为最大化期望收益,同时利用二阶随机占优约束度量风险,满足期望收益二阶随机占优预定的参考目标收益。与传统的二阶随机占优投资组合优化模型不同,本文考虑不确定的投资收益率,并未知其精确的概率分布,但属于某一不确定集合,建立鲁棒二阶随机占优投资组合优化模型,借助鲁棒优化理论,推导出对应的鲁棒等价问题。最后,采用S&P 500股票市场的实际数据,对模型进行不同训练样本规模和不确定集合下的最优投资组合的权重、样本内和样本外不确定参数对期望收益的影响的分析。结果表明,投资收益率在最新的历史数据规模下得出的投资策略,能够获得较高的样本外期望收益,对未来投资更具参考意义。在保证样本内解的最优性的同时,也能取得较高的样本外期望收益和随机占优约束被满足的可行性。     

Robust ranking and portfolio optimization

The portfolio optimization problem has attracted researchers from many disciplines to resolve the issue of poor out-of-sample performance due to estimation errors in the expected returns. A practical method for portfolio construction is to use assets’ ordering information, expressed in the form of preferences over the stocks, instead of the exact expected returns. Due to the fact that the ranking itself is often described with uncertainty, we introduce a generic robust ranking model and apply it to portfolio optimization. In this problem, there are n objects whose ranking is in a discrete uncertainty set. We want to find a weight vector that maximizes some generic objective function for the worst realization of the ranking. This robust ranking problem is a mixed integer minimax problem and is very difficult to solve in general. To solve this robust ranking problem, we apply the constraint generation method, where constraints are efficiently generated by solving a network flow problem. For empirical tests, we use post-earnings-announcement drifts to obtain ranking uncertainty sets for the stocks in the DJIA index. We demonstrate that our robust portfolios produce smaller risk compared to their non-robust counterparts.     

Adaptive and robust radiation therapy optimization for lung cancer

A previous approach to robust intensity-modulated radiation therapy (IMRT) treatment planning for moving tumors in the lung involves solving a single planning problem before the start of treatment and using the resulting solution in all of the subsequent treatment sessions. In this paper, we develop an adaptive robust optimization approach to IMRT treatment planning for lung cancer, where information gathered in prior treatment sessions is used to update the uncertainty set and guide the reoptimization of the treatment for the next session. Such an approach allows for the estimate of the uncertain effect to improve as the treatment goes on and represents a generalization of existing robust optimization and adaptive radiation therapy methodologies. Our method is computationally tractable, as it involves solving a sequence of linear optimization problems. We present computational results for a lung cancer patient case and show that using our adaptive robust method, it is possible to attain an improvement over the traditional robust approach in both tumor coverage and organ sparing simultaneously. We also prove that under certain conditions our adaptive robust method is asymptotically optimal, which provides insight into the performance observed in our computational study. The essence of our method – solving a sequence of single-stage robust optimization problems, with the uncertainty set updated each time – can potentially be applied to other problems that involve multi-stage decisions to be made under uncertainty.     

Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

This paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights. The notion of a deviation interval is introduced, which allows us to characterize the optimality and the robustness of solutions and elements. The problem of computing deviation intervals is addressed and some new complexity results in this field are provided. Possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones.     

Adjustable Robust Optimization Models for a Nonlinear Two-Period System

We study two-period nonlinear optimization problems whose parameters are uncertain. We assume that uncertain parameters are revealed in stages and model them using the adjustable robust optimization approach. For problems with polytopic uncertainty, we show that quasiconvexity of the optimal value function of certain subproblems is sufficient for the reducibility of the resulting robust optimization problem to a single-level deterministic problem. We relate this sufficient condition to the cone-quasiconvexity of the feasible set mapping for adjustable variables and present several examples and applications satisfying these conditions. This work was partially supported by the National Science Foundation, Grants CCR-9875559 and DMS-0139911, and by Grant-in-Aid for Scientific Research from the Ministry of Education, Sports, Science and Culture of Japan, Grant 16710110.     

Wait-and-judge scenario optimization

We consider convex optimization problems with uncertain, probabilistically described, constraints. In this context, scenario optimization is a well recognized methodology where a sample of the constraints is used to describe uncertainty. One says that the scenario solution generalizes well, or has a high robustness level, if it satisfies most of the other constraints besides those in the sample. Over the past 10 years, the main theoretical investigations on the scenario approach have related the robustness level of the scenario solution to the number of optimization variables. This paper breaks into the new paradigm that the robustness level is a-posteriori evaluated after the solution is computed and the actual number of the so-called support constraints is assessed (wait-and-judge). A new theory is presented which shows that a-posteriori observing k support constraints in dimension \(d > k\) allows one to draw conclusions close to those obtainable when the problem is from the outset in dimension k. This new theory provides evaluations of the robustness that largely outperform those carried out based on the number of optimization variables.     

A note on issues of over-conservatism in robust optimization with cost uncertainty

We identify and discuss issues of hidden over-conservatism in robust linear optimization, when the uncertainty set is polyhedral with a budget of uncertainty constraint. The decision-maker selects the budget of uncertainty to reflect his degree of risk aversion, i.e. the maximum number of uncertain parameters that can take their worst-case value. In the first setting, the cost coefficients of the linear programming problem are uncertain, as is the case in portfolio management with random stock returns. We provide an example where, for moderate values of the budget, the optimal solution becomes independent of the nominal values of the parameters, i.e. is completely disconnected from its nominal counterpart, and discuss why this happens. The second setting focusses on linear optimization with uncertain upper bounds on the decision variables, which has applications in revenue management with uncertain demand and can be rewritten as a piecewise linear problem with cost uncertainty. We show in an example that it is possible to have more demand parameters equal their worst-case value than what is allowed by the budget of uncertainty, although the robust formulation is correct. We explain this apparent paradox.     

Minmax robustness for multi-objective optimization problems

In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.     

Uncertain multiobjective traveling salesman problem

Traveling salesman problem is a fundamental combinatorial optimization model studied in the operations research community for nearly half a century, yet there is surprisingly little literature that addresses uncertainty and multiple objectives in it. A novel TSP variation, called uncertain multiobjective TSP (UMTSP) with uncertain variables on the arc, is proposed in this paper on the basis of uncertainty theory, and a new solution approach named uncertain approach is applied to obtain Pareto efficient route in UMTSP. Considering the uncertain and combinatorial nature of UMTSP, a new ABC algorithm inserted with reverse operator, crossover operator and mutation operator is designed to this problem, which outperforms other algorithms through the performance comparison on three benchmark TSPs. Finally, a new benchmark UMTSP case study is presented to illustrate the construction and solution of UMTSP, which shows that the optimal route in deterministic TSP can be a poor route in UMTSP.     

Short sales in Log-robust portfolio management

This paper extends the Log-robust portfolio management approach to the case with short sales, i.e., the case where the manager can sell shares he does not yet own. We model the continuously compounded rates of return, which have been established in the literature as the true drivers of uncertainty, as uncertain parameters belonging to polyhedral uncertainty sets, and maximize the worst-case portfolio wealth over that set in a one-period setting. The degree of the manager’s aversion to ambiguity is incorporated through a single, intuitive parameter, which determines the size of the uncertainty set. The presence of short-selling requires the development of problem-specific techniques, because the optimization problem is not convex. In the case where assets are independent, we show that the robust optimization problem can be solved exactly as a series of linear programming problems; as a result, the approach remains tractable for large numbers of assets. We also provide insights into the structure of the optimal solution. In the case of correlated assets, we develop and test a heuristic where correlation is maintained only between assets invested in. In computational experiments, the proposed approach exhibits superior performance to that of the traditional robust approach.     

New models for the robust shortest path problem: complexity, resolution and generalization

In optimization, it is common to deal with uncertain and inaccurate factors which make it difficult to assign a single value to each parameter in the model. It may be more suitable to assign a set of values to each uncertain parameter. A scenario is defined as a realization of the uncertain parameters. In this context, a robust solution has to be as good as possible on a majority of scenarios and never be too bad. Such characterization admits numerous possible interpretations and therefore gives rise to various approaches of robustness. These approaches differ from each other depending on models used to represent uncertain factors, on methodology used to measure robustness, and finally on analysis and design of solution methods. In this paper, we focus on the application of a recent criterion for the shortest path problem with uncertain arc lengths. We first present two usual uncertainty models: the interval model and the discrete scenario set model. For each model, we then apply a criterion, called bw-robustness (originally proposed by B. Roy) which defines a new measure of robustness. According to each uncertainty model, we propose a formulation in terms of large scale integer linear program. Furthermore, we analyze the theoretical complexity of the resulting problems. Our computational experiments perform on a set of large scale graphs. By observing the results, we can conclude that the approved solvers, e.g. Cplex, are able to solve the mathematical models proposed which are promising for robustness analysis. In the end, we show that our formulations can be applied to the general linear program in which the objective function includes uncertain coefficients.     

Robust economic order quantity models

In this paper, we study the classical economic order quantity (EOQ) model under significant. In particular, the problem under consideration is the economic order quantity model with the input data of the demand rate, the order cost, and the holding cost rate being uncertain. A robustness approach based on scenario characterization of the input data is adopted to describe the uncertainties. The aim of the approach is to find an inventory policy that performs well under all realizable input data scenarios. An efficient linear time algorithm is devised to find the robust decisions. Analytical results are obtained for the case where input data are defined in continuous intervals. Comparisons on performances between the robust decisions and the stochastic optimization decisions are conducted. The results demonstrate the advantages of robustness approach.     

Robust optimization and portfolio selection: The cost of robustness

Robust optimization is a tractable alternative to stochastic programming particularly suited for problems in which parameter values are unknown, variable and their distributions are uncertain. We evaluate the cost of robustness for the robust counterpart to the maximum return portfolio optimization problem. The uncertainty of asset returns is modelled by polyhedral uncertainty sets as opposed to the earlier proposed ellipsoidal sets. We derive the robust model from a min-regret perspective and examine the properties of robust models with respect to portfolio composition. We investigate the effect of different definitions of the bounds on the uncertainty sets and show that robust models yield well diversified portfolios, in terms of the number of assets and asset weights.     

Robust Optimization for Power Systems Capacity Expansion under Uncertainty

We develop a robust optimization model for planning power system capacity expansion in the face of uncertain power demand. The model generates capacity expansion plans that are both solution and model robust. That is, the optimal solution from the model is ‘almost’ optimal for any realization of the demand scenarios (i.e. solution robustness). Furthermore, the optimal solution has reduced excess capacity for any realization of the scenarios (i.e. model robustness). Experience with a characteristic test problem illustrates not only the unavoidable trade-offs between solution and model robustness, but also the effectiveness of the model in controlling the sensitivity of its solution to the uncertain input data. The experiments also illustrate the differences of robust optimization from the classical stochastic programming formulation.     

Two-stage network constrained robust unit commitment problem

For a current deregulated power system, a large amount of operating reserve is often required to maintain the reliability of the power system using traditional approaches. In this paper, we propose a two-stage robust optimization model to address the network constrained unit commitment problem under uncertainty. In our approach, uncertain problem parameters are assumed to be within a given uncertainty set. We study cases with and without transmission capacity and ramp-rate limits (The latter case was described in Zhang and Guan (2009), for which the analysis part is included in Section 3 in this paper). We also analyze solution schemes to solve each problem that include an exact solution approach and an efficient heuristic approach that provides tight lower and upper bounds for the general network constrained robust unit commitment problem. The final computational experiments on an IEEE 118-bus system verify the effectiveness of our approaches, as compared to the nominal model without considering the uncertainty.