Uncertain Optimization Problems Research Articles

Handling Optimization under Uncertainty Problem Using Robust Counterpart Methodology

In this paper we discuss the robust counterpart (RC) methodology to handle the optimization under uncertainty problem as proposed by Ben-Tal and Nemirovskii. This optimization methodology incorporates the uncertain data in U a so-called uncertainty set and replaces the uncertain problem by its so-called robust counterpart. We apply the RC approach to uncertain Conic Optimization (CO) problems, with special attention to robust linear optimization (RLO) problem and include a discussion on parametric uncertainty for that case. Some new supported examples are presented to give a clear description of the used of RC methodology theorem.

A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals

We show that many different concepts of robustness and of stochastic programming can be described as special cases of a general non-linear scalarization method by choosing the involved parameters and sets appropriately. This leads to a unifying concept which can be used to handle robust and stochastic optimization problems. Furthermore, we introduce multiple objective (deterministic) counterparts for uncertain optimization problems and discuss their relations to well-known scalar robust optimization problems by using the non-linear scalarization concept. Finally, we mention some relations between robustness and coherent risk measures.

Worst‐case regret algorithms for selected optimization problems with interval uncertainty

PurposeThe purpose of this paper is to consider selected optimization problems with parameter uncertainty. A case is studied when uncertain parameters in functions undergoing the optimization belong to intervals of known bounds as well as the absolute regret based approach for coping with such an uncertainty is applied. The paper presents three different cases depending on properties of optimization problems and proposes which method can be used to solve corresponding problems.Design/methodology/approachThe worst‐case absolute regret method is employed to manage interval uncertainty in functions to be optimized. To solve resulting uncertain optimization problems, optimal, approximate as well as heuristic solution algorithms have been elaborated for particular problems presented and described in the paper. The latter one is based on Scatter Search metaheuristics.FindingsSolution algorithms for worst‐case absolute regret versions of the following optimization problems have been determined: resource allocation in a complex of independent operations and two task scheduling problems Q‖∑Ci and P‖Cmax.Research limitations/implicationsIt is very difficult to generalize the results obtained and to use them for solving other optimization problems which correspond to real‐world applications. Such new cases require separate investigations.Practical implicationsThe considered allocations as well as task scheduling problems have plenty of applications, for example in computer and manufacturing systems. Their versions with not precisely known parameters can be met commonly.Originality/valueThe investigations presented correspond to previous works on so‐called minimax regret problems and extend them for new optimization problems.

Randomized Solutions to Convex Programs with Multiple Chance Constraints

The scenario-based optimization approach (`scenario approach') provides an intuitive way of approximating the solution to chance-constrained optimization programs, based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty (`scenarios'). A key merit of this approach is that it neither assumes knowledge of the uncertainty set, as it is common in robust optimization, nor of its probability distribution, as it is usually required in stochastic optimization. Moreover, the scenario approach is computationally efficient as its solution is based on a deterministic optimization program that is canonically convex, even when the original chance-constrained problem is not. Recently, researchers have obtained theoretical foundations for the scenario approach, providing a direct link between the number of scenarios and bounds on the constraint violation probability. These bounds are tight in the general case of an uncertain optimization problem with a single chance constraint. However, this paper shows that these bounds can be improved in situations where the constraints have a limited `support rank', a new concept that is introduced for the first time. This property is typically found in a large number of practical applications---most importantly, if the problem originally contains multiple chance constraints (e.g. multi-stage uncertain decision problems), or if a chance constraint belongs to a special class of constraints (e.g. linear or quadratic constraints). In these cases the quality of the scenario solution is improved while the same bound on the constraint violation probability is maintained, and also the computational complexity is reduced.

Modeling and Solving Production Planning Problem under Uncertainty: A Case Study

Data in many real life engineering and economical problems suffer from inexactness. In the real world there are many forms of uncertainty that affect production processes. Uncertainty always exists in practical engineering problems. in order to deal with the uncertain optimization problems, fuzzy and stochastic approaches are commonly used to describe the imprecise characteristics. Herein we assume some intervals in which the data can simultaneously and independently perturb. In this study production planning related data of Al- Araby firm for electric sets in Egypt was collected. A production planning model based on linear programming (LP) was formulated. This formulation based on the outcomes of collected data. The data includes the amount of required and available resources, the demand, the cost of production, the cost of unmet demand, the cost of inventory holding and the revenue. in this work, the objective is to maximize the revenues net of the production, inventory and lost sales costs. The general LP model was solved by using software named Win QSB.

An efficient interval computing technique for bound-constrained uncertain optimization problems

In this article, a competent interval-oriented approach is proposed to solve bound-constrained uncertain optimization problems. This new class of problems is considered here as an extension of the classical bound-constrained optimization problems in an inexact environment. The proposed technique is nothing but an imitation of the well-known interval analysis-based branch-and-bound optimization approach. Efficiency of this technique is strongly dependent on division, bounding, selection/rejection and termination criteria. The technique involves a multisection division criterion of the accepted/proposed search region. Then, we have employed the interval-ranking definitions with respect to the pessimistic decision makers’ point of view given by Mahato and Bhunia [Interval-arithmetic-oriented interval computing technique for global optimization, Appl. Math. Res. Express 2006 (2006), pp. 1–19] to compare the interval-valued objectives calculated in each subregion and also to select the subregion containing the best interval objective value. The process is continued until the interval width for each variable in the accepted subregion is negligible and ultimately the global or close-to-global interval-valued optimal solution is obtained. The proposed technique has been evaluated numerically using a wide set of newly introduced univariate/multivariate test problems. Finally, to compare the computational results obtained by the proposed method, the graphical representation for some test problems is given.

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.

Stability analysis and regularization of uncertain linear multi-objective integer optimization problems

This article presents a systematic approach to analysing linear integer multi-objective optimization problems with uncertainty in the input data. The goal of this approach is to provide decision makers with meaningful information to facilitate the selection of a solution that meets performance expectations and is robust to input parameter uncertainty. Standard regularization techniques often deal with global stability concepts. The concept presented here is based on local quasi-stability and includes a local regularization technique that may be used to increase the robustness of any given efficient solution or to transform efficient solutions that are not robust (i.e. unstable), into robust solutions. An application to a multi-objective problem drawn from the mining industry is also presented.

A solution procedure for estimating the scaling constants of the additive utility function under a fuzzy environment

Data in many real life engineering and economic problems suffer from inexactness. In the real world there are many forms of uncertainty that affect the utility function. Uncertainty always exists in practical engineering problems. To deal with uncertain optimization problems, fuzzy, stochastic and interval numbers approaches are commonly used to describe the imprecise characteristics. Herein, we consider the estimation of the scaling constants of the additive utility function under a fuzzy environment. The investigated procedure uses the Kuhn-Tucker necessary optimality conditions. An illustrative numerical example is given to clarify the developed procedure.

On the application of uncertainty models in copying machine maintenance

A maintenance system is necessary to maintain an efficient production system and reduce the possibility of work being suspended due to machine breakdown. For every manufacturing company the objective is to produce goods at a profit and this is only achieved using an effective maintenance system. Maintenance is required for all types of machinery. The copying machine is one of the most important inventions of the 20th century. In this paper, the maintenance of copying machines is focused on. For this purpose, a multi-objective nonlinear programming model with uncertainty is introduced. Data in many real life engineering and economic problems suffers from inexactness. And uncertainty also exists. In order to deal with uncertain optimization problems, fuzzy and stochastic approaches are commonly used to describe these imprecise characteristics. Here three different approaches to uncertainty are compare; the fuzzy programming approach, the interval number programming approach and the stochastic programming approach. A numerical example is introduced to clarify the proposed method.

Automatic robust convex programming

This paper presents the robust optimization framework in the modelling language YALMIP, which carries out robust modelling and uncertainty elimination automatically and allows the user to concentrate on the high-level model. While introducing the software package, a brief summary of robust optimization is given, as well as some comments on modelling and tractability of complex convex uncertain optimization problems.

Reliability-Based Multiobjective Design Optimization under Interval Uncertainty

This paper studies the reliability-based multiobjective optimization by using a new interval strategy to model uncertain parameters. A new satisfac- tion degree of interval, which is significantly extended from (0, 1) to (-¥, +¥), is introduced into the non-probabilistic reliability-based optimization. Based on a predefined satisfaction degree level, the uncertain constraints can be effectively transformed into deterministic ones. The interval number programming method is applied to change each uncertain objective function to a deterministic two-objective optimization. So in this way the uncertain multiobjective optimization problem is transformed into a deterministic optimization problem and a reliability-based multi- objective optimization is then established. For sophisticated engineering problems, the objectives and constraints are modeled by using the response surface (RS) ap- proximation method to improve the optimization efficiency. Thus the reliability- based multiobjective optimization is combined with the RS approximation models to form an approximation optimization problem. For the multiobjective optimiza- tion, the Pareto sets can be obtained with different satisfactory degree levels. Two numerical examples and one real-world crashworthiness design for vehicle frontal structure are presented to demonstrate the effectiveness of the proposed approach.

Differential Evolution: A Survey of the State-of-the-Art

Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms in current use. DE operates through similar computational steps as employed by a standard evolutionary algorithm (EA). However, unlike traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used for generating the offspring. Since its inception in 1995, DE has drawn the attention of many researchers all over the world resulting in a lot of variants of the basic algorithm with improved performance. This paper presents a detailed review of the basic concepts of DE and a survey of its major variants, its application to multiobjective, constrained, large scale, and uncertain optimization problems, and the theoretical studies conducted on DE so far. Also, it provides an overview of the significant engineering applications that have benefited from the powerful nature of DE.

A relaxation algorithm with a probabilistic guarantee for robust deviation optimization

Three measures of robustness (absolute robustness, deviation robustness and relative robustness), whose choice depends on the goals of the decision maker, have been proposed for uncertain optimization problems. Absolute robustness has been thoroughly studied, whereas the others have been studied to less of a degree. We focus on deviation robustness for uncertain convex quadratic programming problems with ellipsoidal uncertainties and propose a relaxation technique based on random sampling for robust deviation optimization problems. We theoretically and experimentally show that solving the relaxation problem gives a tighter lower bound than solving a simple sampled relaxation problem. Furthermore, we measure the robustness of the solution in a probabilistic setting. The number of random samples is estimated for obtaining an approximate solution with a probabilistic guarantee, and the approximation error is evaluated a-priori and a-posteriori. Our relaxation algorithm with a probabilistic guarantee utilizes a-posteriori assessment to evaluate the accuracy of the approximate solutions.

A nonlinear interval number programming method for uncertain optimization problems

In this paper, a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, in which the robustness of design is considered. Through a modified possibility degree, the uncertain inequality and equality constraints are changed to deterministic inequality constraints. The two objective functions are converted into a single-objective problem through the linear combination method, and the deterministic inequality constraints are treated with the penalty function method. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimization problem. Two numerical examples are investigated to demonstrate the effectiveness of the present method.

Enterprise resource planning implementation decision & optimization models

To study the uncertain optimization problems on implementation schedule, time-cost trade-off and quality in enterprise resource planning (ERP) implementation, combined with program evaluation and review technique (PERT), some optimization models are proposed, which include the implementation schedule model, the timecost trade-off model, the quality model, and the implementation time-cost-quality synthetic optimization model. A PERT-embedded genetic algorithm (GA) based on stochastic simulation technique is introduced to the optimization models solution. Finally, an example is presented to show that the models and algorithm are reasonable and effective, which can offer a reliable quantitative decision method for ERP implementation.

A sequential nonlinear interval number programming method for uncertain structures

A sequential nonlinear interval number programming (SNINP) method is suggested to deal with the uncertain optimization problems. A general uncertain optimization model is investigated in which the objective function and constraints are both nonlinear and uncertain. A nonlinear interval number programming (NINP) method is employed to transform the uncertain optimization problem into a deterministic two-objective optimization problem. Then using the linear combination method and the constraint penalty function method, a deterministic single-objective and non-constraint optimization problem is formulated in terms of a penalty function. Combining this NINP method with a modified approximation management framework (AMF), an efficient SNINP method is then constructed. At each iterative step, an approximation optimization problem is created based on the Latin Hypercube Design (LHD) and the quadratic polynomial response surface approximation (RSA), and it can then be solved by the NINP method efficiently. The trust region method is used to manage the sequence of the approximation optimization problems based on a reliability index. An efficient method is suggested to calculate the actual penalty function and whereby the reliability index, and based on it the current design space can be updated. Two numerical examples are presented to demonstrate the effectiveness of the present method.

Uncertain optimization of composite laminated plates using a nonlinear interval number programming method

An uncertain optimization method is suggested to optimize the ply orientations of a composite laminated plate with uncertain material property to obtain a maximum stiffness in the thickness direction. Interval numbers are employed to model the uncertainty of the material property by which only the bounds of the uncertain parameters are needed. An uncertain optimization problem is created based on the elastic waves which are computed by a hybrid numerical method efficiently. A nonlinear interval number programming method is suggested to deal with the uncertain optimization problem. In this method a new order relation of interval number is constructed to transform the uncertain objective function into a deterministic two-objective optimization problem, and a hybrid optimization algorithm is used to eliminate the optimization nesting of the simulation model and whereby improve the computation efficiency. Three numerical examples are investigated, and the optimization results demonstrate the effectiveness of the present method.

Modeling and solving uncertain optimization problems in YALMIP

A considerable amount of optimization problems arising in the control and systems theory field can be seen as special instances of robust optimization. Much of the modeling effort in these cases is spent on converting an uncertain problem to a robust counterpart without uncertainty. Since many of these conversions follow standard procedures, it is amenable to software support. This paper presents the robust optimization framework in the modeling language YALMIP, which carries out the uncertainty elimination automatically, and allows the user to concentrate on the high-level model instead.

An uncertain structural optimization method based on nonlinear interval number programming and interval analysis method

An optimization method is proposed to solve uncertain structural problems based on a nonlinear interval number programming method and an interval analysis method. A nonlinear interval number programming method is suggested to transform the uncertain optimization problem to a deterministic multi-objective optimization problem based on an order relation of interval. For each specific design vector, an interval analysis method is applied to calculate the interval of the objective function caused by uncertainty, and whereby the optimization nesting problem can be solved. A non-constraint and single-objective optimization problem is then formulated through the linear combination method of multi-objective optimization and the penalty function method. An intergeneration projection genetic algorithm is employed to seek for Pareto optimum of the uncertain problem. The presented method is applied to a benchmark test of ten-bar truss and a practical automobile frame, and the optimization results demonstrate its efficiency.