Uncertain Multi-objective Optimization Research Articles

Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set

In this paper, a steepest descent method is developed for the robust counterpart of uncertain multiobjective optimization problems under finite uncertainty. The robust counterpart under consideration is the minimum of objective wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. To solve this robust counterpart with the help of the steepest descent method, a subproblem is constructed and solved to find a descent direction. An Armijo-type inexact line search technique is employed to find a suitable step length. With the help of the descent direction and step length, a sequence is generated. The convergence of the proposed method is justified under some common assumptions. Finally, the algorithm is verified and compared with one existing method with the help of some numerical problems.

A modified quasi-Newton method for uncertain multiobjective optimization problems under a finite uncertainty set

ABSTRACT In this article, a modified quasi-Newton method is developed for the robust counterpart (RC) of an uncertain multiobjective optimization problem (UMOP). Specifically, the RC of a UMOP is the minimum of an objective-wise worst-case (OWWC) type RC. To determine the descent direction for the RC, a subproblem using a common Hessian approximation is constructed. An inexact line search technique based on the Armijo technique is employed to find an appropriate step length. Furthermore, a modified version of the Davidon–Fletcher–Powell (DFP) update formula is developed for the RC. A modified quasi-Newton algorithm is built using descent direction step length, and DFP Hessian update formula. Furthermore, the convergence of the modified quasi-Newton algorithm is also discussed. Finally, the modified quasi-Newton algorithm is illustrated with a set of test problems and compared with the weighted sum method. Moreover, it is demonstrated that the algorithm is effective for both convex and non-convex problems.

Robust optimality and duality for composite uncertain multiobjective optimization in Asplund spaces with its applications

Robust optimality and duality for composite uncertain multiobjective optimization in Asplund spaces with its applications

Optimality Conditions in Uncertain Multi-Objective Optimization Problems with a Variable Domination Structure

In this paper, we first establish some new properties of a kind of nonlinear scalarization function. Second, based on set approaches, we introduce a class of new definitions of robustness for an uncertain multi-objective optimization problem with a variable domination structure. Finally, by applying the scalarization method, we obtain the necessary and sufficient optimality conditions of the robustness for the uncertain multi-objective optimization problem with a variable domination structure in a more general setting. Some of the obtained results extend and imply the corresponding ones in recent literature.

Uncertain optimal attitude control for space power satellite based on interval Riccati equation with non-probabilistic time-dependent reliability

Considering the multi-source uncertainties and disturbances in the attitude dynamics of space power satellite (SPS), an interval dynamic model of SPS is proposed based on the non-probabilistic theory, which is controlled by a novel uncertainty-based linear–quadratic regulator (LQR) method with time-dependent reliability. Based on the interval analysis method, an interval time-varying attitude dynamic model of SPS for sun-pointing is established considering the uncertain inertia and initial states as unknown-but-bounded parameters. Feedforward and feedback controllers are designed to offset the disturbance and control attitude, respectively. An interval Riccati equation-based optimal control method is proposed to develop a novel uncertainty-based LQR method. Non-probabilistic time-dependent reliability is constituted based on the interval process and first passage theory to assess attitude states. An uncertain multi-objective optimization of attitude control with reliability constraints is proposed to realize the uncertain LQR method. Verified by a numerical example of SPS, the proposed method takes only 0.6‰∼0.8‰ time consumption of 10,000 Monte Carlo simulations while maintaining a certain accuracy.

The point-based robustness gap for uncertain multiobjective optimization

In robust single-objective optimization, the robustness gap is a measure of the distance between the robust optimal objective value and the optimal objective values of the scenarios. While robust multiobjective optimization is a growing field of study, no notion of a robustness gap has been proposed. A concept of a point-based robustness gap for uncertain multiobjective optimization problems is introduced. The gap is defined as the minimal distance between the robust Pareto set and the Pareto sets of the scenarios. It is shown that the gap is zero whenever the uncertainty is constraint-wise and objective-wise, supplementing a major result about the single-objective robustness gap. Because the distance between Pareto sets is hard to compute, lower and upper bounds on the gap are constructed for convex problems. Specific results about the zero gap and the bounds are presented for linear problems. Numerical examples are included.

Newton’s method for uncertain multiobjective optimization problems under finite uncertainty sets

Newton’s method for uncertain multiobjective optimization problems under finite uncertainty sets

Unified Robust Necessary Optimality Conditions for Nonconvex Nonsmooth Uncertain Multiobjective Optimization.

This paper is concerned with nonconvex nonsmooth uncertain multiobjective optimization problems, in which the decision variable of both objective and constraint functions is defined on Banach space while uncertain parameters are defined on arbitrary nonempty (may not be compact) sets. We employ the Stone–C̆ech compactification of uncertainty sets and the upper semicontinuous regularization of original functions with respect to uncertain parameters, giving rise to unified robust necessary optimality conditions for the local robust weakly efficient solution of the considered problem. Moreover, we derive weak and strong KKT robust necessary conditions via the constraint qualification and the regularity condition, respectively. Several examples are provided to illustrate the validity of our results.

Toward a sustainable economic production quantity model employing triple bottom line strategy: Uncertain multi-objective optimization with lost sales and full back-order

Toward a sustainable economic production quantity model employing triple bottom line strategy: Uncertain multi-objective optimization with lost sales and full back-order

Multiobjective Optimization of Process Parameters of Silicon Single Crystal Growth

In order to obtain higher quality silicon single crystal, a hybrid strategy for modeling of Czochralski silicon crystal growth and optimizing of process parameters is presented in the paper. The hybrid strategy includes the computational fluid dynamics (CFD) method, neural network of group method of data handing (GMDH), and improved nondominated sorting genetic algorithm II (NSGA-II). The shape variable of solid-liquid interface h and the defect evaluation criteria V / G are set to objective functions according to engineering experience and process requirement. The polynomial of the objective functions is produced by GMDH and CFD. Ultimately, an improved elitist strategy and crowding distance NSGA-II is proposed in the paper to obtain the Pareto optimum solution by the objective functions identified by the GMDH. Compared with other optimization algorithms, the improved NSGA-II can increase the lateral diversity and the uniform distribution of the nondominated solutions. Engineering validation proved that the proposed hybrid strategy can effectively solve the complex uncertain multiobjective optimization problem of system and provides a new method for obtaining high-quality crystal growth process parameters.

The Lightly Robust Max-Ordering Solution Concept for Uncertain Multiobjective Optimization Problems: An Ambulance Location Problem with Unavailability

This study introduces a robust concept for considering uncertain multiobjective optimization problems, called the lightly robust max-ordering solution. This introduced solution concept offers the best option for solving issues based on the maximum cost in the worst-case scenario. Introducing a tolerable relaxation parameter can be used to increase the robustness of the solution but, at the same time, the desirable property of such a solution with respect to the nominal scenario might be decreased. Subsequently, the two measurements, which are the ‘gain in robustness’ and the ‘price to be paid for robustness’, are considered. These measurements are needed to support a decision maker to find more desirable lightly robust max-ordering solutions with a beneficial trade-off between the robustness of solutions and the quality of solutions in an undisturbed situation. Moreover, an algorithm for finding the proposed solution is presented and discussed. An instance of the benefits of the suggested solution concept is used on an example of ambulance location planning if ambulances may be unavailable.

Optimization path design for urban travel system based on CO2-congestion-satisfaction multi-objective synergy: Case study in Suzhou, China

The optimization of urban travel system can effectively reduce CO2 emissions and alleviate road congestion. It can also improve resident satisfaction. Thus, a multi-objective optimization model from the government-resident perspective is developed to coordinate the needs of both sides for urban travel system. Meanwhile, a two-step uncertain multi-objective optimization algorithm combining interval-value interaction algorithm and non-dominated ranking genetic algorithm is designed to solve the model. Results showed that in the scenario based on the travel conditions of Suzhou in 2019, shifting the travel mode from Private car, Bike/e-bike to Bus, Metro is the optimal solution. The value of CO2 emissions, Road congestion and Satisfaction are reduced by (-10.63%, -9.66%, -3.15%) in the best case and (-1.56%, -2.72%, -4.33%) in the worst case. To further analyze the effect of policy implementation on optimization results, management group (MG), technology group (TG) and resource group (RG) policies are introduced into the original model. It is found that the average optimization effect of MG is the best, but different policies have emphases on different objectives. According to the optimization features of different policies, a policy implementation path is proposed: MG→TG→RG. The above conclusions can provide theoretical guidance for the optimization of urban travel system.

Robust nonsmooth optimality conditions for multiobjective optimization problems with infinitely many uncertain constraints

This paper concentrates on uncertain multiobjective optimization problems with an arbitrary number of uncertain constraints under nonconvex and nonsmooth assumptions. We analyse the properties of robust infinite constraints and arrive at the Clarke subdifferential of the double supremum function. Subsequently, we employ the obtained subdifferential rules to study KKT robust necessary and sufficient optimality conditions for two types of uncertain multiobjective optimization problems. Simultaneously, some illustrative examples are provided to show the validity of the main results. Our results are new and generalize several corresponding results in the literature.

LR-NIMBUS: an interactive algorithm for uncertain multiobjective optimization with lightly robust efficient solutions

LR-NIMBUS: an interactive algorithm for uncertain multiobjective optimization with lightly robust efficient solutions

Characterizations of Approximate Optimality Conditions for Fractional Semi-Infinite Optimization Problems With Uncertainty

This paper is devoted to the investigation of a class of multi-objective fractional semi-infinite optimization problems with uncertain data. Firstly, a robust counterpart for the uncertain multi-objective optimization problem is introduced in terms of robust optimization method. And following the idea due to Dinkelbach, we associate the robust counterpart with a multi-objective optimization problem. Then, by using a scalarization method, its corresponding scalarization optimization problem is established. The relationship between robust solutions of the multi-objective optimization problem and that of its corresponding scalarization optimization problem is described. Finally, by using a robust type sub-differential constraint qualification, the robust optimality condition for approximate quasi-efficient solutions of the multi-objective fractional optimization problem is established.

Optimality conditions for robust weak sharp efficient solutions of nonsmooth uncertain multiobjective optimization problems

In this paper, we investigate an uncertain multiobjective optimization problem involving nonsmooth and nonconvex functions. The notion of a (local/global) robust weak sharp efficient solution is introduced. Then, we establish necessary and sufficient optimality conditions for local and/or the robust weak sharp efficient solutions of the considered problem. These optimality conditions are presented in terms of multipliers and Mordukhovich/limiting subdifferentials of the related functions.

On the properties of lexicographic tolerable robust solution sets for uncertain multi-objective optimization problems

This study provides the important properties of the lexicographic tolerable robust solution for uncertain multi-objective optimization problems which was introduced by Boriwan et al. [Boriwan, P.; Ehrgott, M.; Kuroiwa, D.; Petrot, N. The lexicographic tolerable robustness concept for uncertain multi-objective optimization problems: a study on water resources management. Sustainability. 12 (2020), no. 18, article number 7582.]. Also, the relationship between the lexicographic tolerable solution concept and the well-known robust solution, as the set-based robust efficiency [Ehrgott, M.; Ide, J.; Schöbel, A. Minmax robustness for multi-objective optimization problems. European Journal of Operational Research. 239 (2014), no. 1, 17-31.], are provided.

On the properties of lexicographic tolerable robust solution sets for uncertain multi-objective optimization problems

This study provides the important properties of the lexicographic tolerable robust solution for uncertain multi-objective optimization problems which was introduced by Boriwan et al. [Boriwan, P.; Ehrgott, M.; Kuroiwa, D.; Petrot, N. The lexicographic tolerable robustness concept for uncertain multi-objective optimization problems: a study on water resources management. Sustainability. 12 (2020), no. 18, article number 7582.]. Also, the relationship between the lexicographic tolerable solution concept and the well-known robust solution, as the set-based robust efficiency [Ehrgott, M.; Ide, J.; Schöbel, A. Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239 (2014), no. 1, 17–31.], are provided.

The price of multiobjective robustness: Analyzing solution sets to uncertain multiobjective problems

Defining and finding robust efficient solutions to uncertain multiobjective optimization problems has been an issue of growing interest recently. Different concepts have been published defining what a “robust efficient” solution is. Each of these concepts leads to a different set of solutions, but it is difficult to visualize and understand the differences between these sets. In this paper we develop an approach for comparing such sets of robust efficient solutions, namely we analyze their outcomes under the nominal scenario and in the worst case using the upper set-less order from set-valued optimization. Analyzing the set of nominal efficient solutions, the set of minmax robust efficient solutions and different sets of lightly robust efficient solutions gives insight into robustness and nominal objective function values of these sets of solutions. Among others we can formally prove that lightly robust efficient solutions are good compromises between nominal efficient solutions and minmax robust efficient solutions. In addition, we also propose a measure to quantify the price of robustness of a single solution. Based on the measure, we propose two strategies which can be used to support a decision maker to find solutions to a multiobjective optimization problem under uncertainty. All our results are illustrated by examples.

The Lexicographic Tolerable Robustness Concept for Uncertain Multi-Objective Optimization Problems: A Study on Water Resources Management

In this study, we introduce a robust solution concept for uncertain multi-objective optimization problems called the lexicographic tolerable robust solution. This approach is advantageous for the practical implementation of problems in which the solution should satisfy priority levels in the objective function and the worst performance vector of the solution obtained by the proposed concept is close to a reference point of the considered problem, within an acceptable tolerance threshold. Important properties of the solution sets of this introduced concept as well as an algorithm for finding such solutions are presented and discussed. We provide the implementation of the proposed lexicographic tolerable robust solution to improve understanding for practitioners by relying on the data of the water resources master plan for Serbia from Simonovic, 2009. Moreover, we are also concerned with the method of updating a desirable solution for fitting with the preferences when compromising of the multiple groups of decision makers is needed.