No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Technical Note--Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming
Abstract
This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {Kj, j = 1, 2, …, n} and a convex resource set K are specified and the feasible region is given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X =\{x \in R^{n}\mid x_{1}K_{1} + x_{2}K_{2} + \cdots + x_{n}K_{n} \subseteq K, x_{j}\geq 0\}$$ \end{document} where the binary operation + refers to addition of sets. The problem is then to find x̄ ∈ X that maximizes the linear function c · x. When the res...
... Moreover, uncertainty is incorporated into the portfolio optimization phase by extending the deterministic EGP models to their robust counterparts in order to ensure model stability in the face of imprecise data. To this end, two RO approaches via a polyhedral uncertainty set (Li et al., 2011) and a combined interval and polyhedral uncertainty set (Bertsimas & Sim, 2004) are adopted and compared with each other for the following reasons: (a) RO via polyhedral uncertainty set guarantees maximum robustness against uncertainty while incurring a high cost, which entices risk averse decision makers; (b) RO via combined interval and polyhedral uncertainty set reaches a compromise between robustness and its cost, appealing to risk-seeking decision makers; (c) RO via the above-mentioned uncertainty sets doesn't contain the overconservatism present in classic worst-case robust models, such as Soyster's (1973) approach, wherein optimality is excessively sacrificed for feasibility. Instead, in the resulting robust counterparts via the aforesaid uncertainty sets, the decision maker has complete control over the degree of conservatism for each constraint with uncertain parameters; and (d) RO via the aforementioned uncertainty sets retains the linearity of the nominal problem, in contrast to certain RO approaches (e.g. ...
... It is worth noting that in the aforesaid robust counterpart formulations, when i is set equal to zero, the constraints are equivalent to that of the nominal problem. Similarly, when i is set equal to |J i |, the robust model acts as conservative as in the robust formulation of Soyster's (1973). ...
This study presents a two-phase approach of Data Envelopment Analysis (DEA) and Goal Programming (GP) for portfolio selection, representing a pioneering attempt at combining these techniques within the context of portfolio selection. The approach expands on the conventional risk and return framework by incorporating additional financial factors and addressing data uncertainty, which allows for a thorough examination of portfolio outcomes while accommodating investor preferences and conservatism levels. The initial phase employs a super-efficiency DEA model to streamline asset selection by identifying suitable investment candidates based on efficiency scores, setting the stage for subsequent portfolio optimization. The second phase leverages the Extended GP (EGP) framework, which facilitates the comprehensive incorporation of investor preferences to determine the optimal weights of the efficient assets previously identified within the portfolio. Each goal is tailored to reflect specific financial factors spanning both technical and fundamental aspects. To tackle data uncertainty, robust optimization is applied. The research contributes to the robust GP (RGP) literature by analyzing new RGP variants, overcoming limitations of traditional and other uncertain GP models by incorporating uncertainty sets. Robust counterparts of the EGP models are accordingly developed using polyhedral and combined interval and polyhedral uncertainty sets, providing a flexible representation of uncertainty in financial markets. Empirical results, based on real data from the Tehran Stock Exchange comprising 779 assets, demonstrate the superiority of the proposed approach over traditional portfolio selection methods across various uncertainty settings. Additionally, a comprehensive sensitivity analysis investigates the impact of uncertainty levels on the robust EGP models. The proposed framework offers guidance to investors and fund managers through a pragmatic approach, enabling informed and robust portfolio decisions by considering efficiency, uncertainty, and extended financial factors.
... Thus, despite being relatively more expensive, the derived robust solution is more reliable and will allow the decision-maker to have a practical solution for several realizations of the uncertain parameters which are difficult to predict. The first robust approach was proposed by Soyster (1973) using a linear optimization model that provides the best feasible solution for all possible realizations of random input data. ...
... El Ghaoui et al. (1998) proposed other approaches which involves solving the robust counterparts of the nominal problem using a quadratic objective function. Although these models can better approximate some types of uncertainties without defaulting to over-conservative solutions, they have the disadvantage of requiring the solution of a non-linear optimization problem, which tends to be more difficult to solve than the linear model of Soyster (1973). As an intermediate strategy to tackle both over-conservatism and high computational effort, Bertsimas and Sim (2004) introduced a methodology to control the conservatism level of the solutions with a linear formulation. ...
An efficient Municipal solid waste (MSW) system is critical to modern cities in order to enhance sustainability and livability of urban life. With this aim, the planning phase of the MSW system should be carefully addressed by decision makers. However, planning success is dependent on many sources of uncertainty that can affect key parameters of the system, e.g., the waste generation rate in an urban area. With this in mind, this paper contributes with a robust optimization model to design the network of collection points (i.e., location and storage capacity), which are the first points of contact with the MSW system. A central feature of the model is a bi-objective function that aims at simultaneously minimizing the network costs of collection points and the required collection frequency to gather the accumulated waste (as a proxy of the collection cost). The value of the model is demonstrated by comparing its solutions with those obtained from its deterministic counterpart over a set of realistic instances considering different scenarios defined by different waste generation rates. The results show that the robust model finds competitive solutions in almost all cases investigated. An additional benefit of the model is that it allows the user to explore trade-offs between the two objectives.
... The concept of static RO was first proposed by Soyster in the 1970s, who studied a linear optimization problem with a box uncertainty set [111]. Later in the 1990s, static RO was formally introduced [15, 16,57] and its computational advantage has resulted in its wide usage in applications, including in portfolio selection [79,121], scheduling [40], operations management [85], etc. ...
... The concept of robust optimization was first introduced by Soyster in 1973, and later developed by El Ghaoui, Ben-Tal, Nemirovski, Bertsimas, and den Hertog [111,13,22]. ...
Robust Optimization (RO) is a popular approach for dealing with uncertain data in optimization. In static robust optimization, decision variables represent here-and-now decisions made without exact knowledge of uncertain parameters but must be feasible when the actual data is within the uncertainty set. However, one mechanism to overcome the limitations of the static RO approach is Adjustable Robust Optimization (ARO), which leverages adaptability. The main difference between static RO and ARO approaches is the decision-making manner. In ARO problems, some variables are here-and-now decisions, while others are wait-and-see decisions made later based on the observed parameters in the uncertainty set.
In this PhD thesis, we address two main topics in mathematical optimization. The first topic concerns a class of nonlinear ARO problems with uncertainty in the objective function and constraints. By utilizing Fenchel’s duality, we derive an equivalent dual reformulation that is a nonlinear static robust optimization problem. We then apply perspective relaxation and an alternating method to handle non-concavity and design a new dual-based cutting plane algorithm that can find a reasonable lower bound for the optimal objective value. Through numerical experiments, we show the effectiveness of the cutting plane algorithm in producing locally robust solutions with an acceptable optimality gap.
The second topic focuses on the reformulation of quadratic optimization problems using ARO. Quadratic Optimization (QO) has been extensively studied in the literature due to its practical applicability in numerous problems. Despite its practicality, QO problems are generally NP-hard. Consequently, researchers have developed various numerical methods for finding approximate optimal solutions. In this thesis, we analyze QO problems through the lens of robust optimization techniques. We first demonstrate that any QO problem can be reformulated as a disjoint bi-convex QO problem. Subsequently, we present an equivalent ARO reformulation and utilize some methods from the relevant literature to approximate this reformulation. Specifically, we show that employing a so-called decision rule technique to approximate the ARO reformulation can be interpreted as applying a linearization-relaxation technique to its bi-convex reformulation problem. Additionally, we have designed an algorithm capable of finding a solution that is close to optimal based on our new reformulations. Our numerical results highlight the efficiency of our algorithm, particularly for large-sized instances, in comparison with standard off-the-shelf solvers. This work offers a novel perspective on quadratic problems and paves the way for further research in this domain of mathematical optimization.
... Since the seminal paper by Soyster [34], robust optimization has become a useful and efficient deterministic mathematical approach to handle problems relating to decision making in the face of data uncertainty [5,7]. Recently, the robust optimization approach has been applied to various domains in multiobjective/vector frameworks with many further developments and high-potential techniques to solve various real-life decision making problems under the data uncertainty (see e.g., [9, 12, 16-18, 20, 23, 24, 26, 36] and the references therein). ...
... , q. As shown in the proof of (ii) of Theorem 1, we show by (34) and (35) that there existû ζ ∈ U ζ , ζ = 1, . . . , m andω l ∈ l , l = 1, . . . ...
This paper studies a class of multiobjective convex polynomial problems, where both the constraint and objective functions involve uncertain parameters that reside in ellipsoidal uncertainty sets. Employing the robust deterministic approach, we provide necessary conditions and sufficient conditions, which are exhibited in relation to second order cone conditions, for robust (weak) Pareto solutions of the uncertain multiobjective optimization problem. A dual multiobjective problem is proposed to examine robust converse, robust weak and robust strong duality relations between the primal and dual problems. Moreover, we establish robust solution relationships between the uncertain multiobjective optimization program and a (scalar) second order cone programming relaxation problem of a corresponding weighted-sum optimization problem. This in particular shows that we can find a robust (weak) Pareto solution of the uncertain multiobjective optimization problem by solving a second order cone programming relaxation.
... The first approach in robust optimization presented by Soyster. He considered a linear programming model in 1973 and tried to counteract to the uncertainty of data, by a conservative approach (Soyster, 1973). The final solution of his approach was far away from optimal solution, because he considered the worst cases. ...
This paper studies the robust vertex centdian location problem on tree networks with interval edge lengths. To obtain a robust solution, we use the minmax regret criterion. First, we obtain atmost n worst case scenario for each pair of vertices on tree networks. Then we reduce this number to logn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \log n $$\end{document}. Finally, the total time for obtaining a robust solution is equal to O(n4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ O(n^{4}) $$\end{document}. This time is reduced to O(n3logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ O(n^{3}\log n) $$\end{document} by using the property of binary search tree and weighted centroid.
... while robust solutions are more resilient, in general, to uncertainty and parameter change, they often have a worse objective function value compared to solutions obtained by standard linear optimization. A study in 2000 propose the following robust problem [91]: ...
This study introduces an integrated method for managing process risks in a Business Process Reengineering (BPR) project, using Robust Data Envelopment Analysis (RDEA) and machine learning (ML). The goal is to prioritize risks based on three standard factors of PFMEA: Severity, Occurrence, and Detection (S-O-D), and incorporating two additional factors (Breakdown Cost and Breakdown Duration) seen as undesirable outputs. The model also accounts for the effect of uncertainty on expert-estimated values by applying disturbance percentages in the linear PFMEA-RDEA model. A machine learning model is proposed to predict new values if partial or total modifications have been made to the processes. The approach was implemented in an au-tomotive sector company, and the results showed the impact of uncertainty on values by com-paring different approaches such as RPN, PFMEA-DEA, and PFMEA-RDEA. A new reduced risk categorization was achieved, who allowed for decision-makers to focus on necessary actions for reengineering.
... Some methods prioritize finding the absolute best solution under the worst-case scenario (highly robust), while others aim for a balance between optimality and robustness. Equation (41) shows the box uncertainty set introduced by Soyster [49]. It is a linear model that prioritizes robustness over optimality and ensures feasibility even under extreme parameter changes. ...
This paper presents an optimal framework for demand response aggregators in wholesale electricity markets. Demand response aggregators provide customers with flexible contracts in the proposed model. These contracts allow for hourly load reductions through load curtailment, load shifting, onsite generation utilization, and energy storage systems. The study suggests the price-based self-scheduling model for demand response aggregators, which seeks to maximize the aggregator's payoff for participation in day-ahead energy markets. The proposed model is implemented on a sample demand response aggregator with uncertainty and without uncertainty. Three methods, box, polyhedral, and ellipsoidal, based on robust optimization, are proposed for uncertainty modeling in this work, and their effectiveness is compared with each other. The results show that if the uncertainty is modeled using the polyhedral method, the value of the objective function will deviate by only 22.72 % compared to the case without uncertainty. However, this deviation in box and ellipsoidal methods is 118.48 % and 49.20 %, respectively, which shows the superiority of the polyhedral method compared to the oval and box methods.
... The concept of robust optimization emerged in the 1970s in the field of linear optimization. A pivotal milestone occurred in 1973 when Soyester devised a pessimistic robust programming method to grapple with imprecise linear programming challenges (Soyster, 1973). The multifaceted nature of robust optimization was delineated by Mulvey et al. (1995) along two dimensions: Feasibility Robustness and Optimality Robustness. ...
Sustainable development in supply chain management is of great importance, as it emphasizes that the procurement process in sustainable supply chains addresses not only current needs but also future requirements. This subject, by improving processes, reducing waste and energy consumption , increasing efficiency, and establishing sustainable relationships with suppliers and customers, contributes to the sustainability and success of companies in international competition. Moreover, hub location is crucial for optimizing distribution processes and reducing costs in the supply chain, especially for perishable products with a defined and limited shelf life, which imposes various constraints on the supply chain. Additionally, to bring mathematical models closer to reality, uncertainty approaches can be employed, and strategies to deal with uncertainty can be considered. This research delves into these issues in the realm of supply chain management, focusing on sustainability and uncertainty, and provides a review of research conducted in the literature of this field.
... The first category of SCP models focuses on the uncertainty of constraint coefficient a i j for i ∈ I , j ∈ J and their existing applications. Initially, Soyster (1973) introduces the uncertainty of the constraint coefficients in linear models. Since there is no guarantee that the classical covering constraint (1b) can be satisfied with a i j uncertainty, the majority of the models reviewed in this section utilize the chance-constrained method for modeling, one of the primary approaches used to solve optimization problems under various uncertainties. ...
The Set Covering Problem (SCP) has been an extensively studied NP-hard problem in the field of combinatorial optimization since 1970. Over the past five decades, a significant amount of research has led to the development of a diverse set of covering models to support decision-making in various areas. However, the SCPs related to real-world applications are often too complex to solve using existing algorithms due to uncertain problem parameters. Thus, given the diversity of new developments, there is a pressing need to know both the current solution approaches and the advanced strategies for studying the uncertain SCP. This study summarizes the various modeling and solution approaches to the SCP when the model parameters are uncertain. Further, this study discusses some promising future research directions of the uncertain SCP that will impact new investigations of decisions on complex and competitive real-world issues.
... Recall that B u denote an uncertain symmetric n × n matrix, b u an uncertain vector and β u is an uncertain scalar. One of the most well-known approaches to study problem (1.2) is robust optimization [4,5,7,8,9,29]. A solution of such an optimization problem is called robust optimal solution. This kind of solution is known to be immunized against uncertainties and noises of a quadratic model. ...
In this work, we study an uncertain quadratic optimization problem by using the concept of stability radius. That allows to prove robustness of uncertain quadratic optimization models. Firstly, we determine the stability radius of this problem by applying Ascoli formula. Secondly, we show that its robust counterpart has at least an optimal solution. Finally, we deal with an application and provide some numerical methods for such an uncertain quadratic optimization problem.
... Given the support w w,t ,w w,t of w w,t and by using Soyster's method [36], (19a) and (19b) can be transformed into the following equivalent problems (20a) and (20b), respectively: ...
Distributionally robust optimization (DRO) has emerged as a favored methodology for addressing the uncertainties stemming from renewable energy sources. However, existing DRO frameworks primarily focus on single types of uncertainty characteristics, such as moments. Exploring novel ambiguity sets that encompass heterogeneous uncertainty information to mitigate decision conservatism is thus an essential and strategic move. This paper introduces a day-ahead optimal scheduling model tailored for electricity-hydrogen systems under renewable uncertainty, with embedded technologies of hydrogen production, storage, and utilization. Three novel ambiguity sets enriched with the moment, Wasserstein distance, and unimodality information are adeptly devised. Building upon these elaborated ambiguity sets, we develop efficient and scalable reformulations of the expected objective function and uncertain constraints, leading to either a tractable mixed-integer second-order cone programming problem or a linear programming problem. We validate the effectiveness and operating flexibility of the proposed electricity-hydrogen model using both a 6-bus test system and the IEEE 118-bus test system. Furthermore, we demonstrate the superior cost performance and computational efficiency of our developed DRO approaches.
... Robust optimization methods are widely used because they do not require specific probability distributions to deal with wind power uncertainty. Since the 1970s, Soyster et al. [11] pioneered the use of linear robust optimization methods for solving uncertain linear programming problems. To overcome the limitations of this approach, a constraint protecting the nominal parameter level was introduced in the literature [12]. ...
This paper presents a bi-level inverse robust economic dispatch optimization model consisting of wind turbines and pumped storage hydropower (PSH). The inner level model aims to minimize the total generation cost, while the outer level introduces the optimal inverse robust index (OIRI) for wind power output based on the ideal perturbation constraints of the objective function. The OIRI represents the maximum distance by which decision variables in the non-dominated frontier can be perturbed. Compared to traditional methods for quantifying the worst-case sensitivity region using polygons and ellipses, the OIRI can more accurately quantify parameter uncertainty. We integrate the grid multi-objective bacterial colony chemotaxis algorithm and the bisection method to solve the proposed model. The former is adopted to solve the inner level problem, while the latter is used to calculate the OIRI. The proposed approach establishes the relationship between the maximum forecast deviation and the minimum generation cost associated with each non-dominated solution in the optimal load allocation. To demonstrate its economic viability and effectiveness, we simulate the proposed approach using real power system operation data and conduct a comparative analysis.
... The pivotal role of robustness is also clear in the context of optimization problems, in which the RO research field is relevant. As a historical note, consider that in 1973, Soyster [35] formulated a linear programming (LP) optimization model, yielding feasible solutions for all data belonging to a convex set. However, this approach leaned towards excessive conservatism, compromising optimality for enhanced robustness. ...
Among the challenges generated by the global climate crisis, a significant concern is the constant increase in energy demand. This leads to the need to ensure that any novel energy systems are not only renewable but also reliable in their performance. A viable solution to increase the available renewable energy mix involves tapping into the potential available in ocean waves and harvesting it via so-called wave energy converters (WECs). In this context, a relevant engineering problem relates to finding WEC design solutions that are not only optimal in terms of energy extraction but also exhibit robust behavior in spite of the harsh marine environment. Indeed, the vast majority of design optimization studies available in the state-of-the-art consider only perfect knowledge of nominal (idealized) conditions, neglecting the impact of uncertainties. This study aims to investigate the information that different robustness metrics can provide to designers regarding optimal WEC design solutions under uncertainty. The applied methodology is based on stochastic uncertainty propagation via a Monte Carlo simulation, exploiting a meta-model to reduce the computational burden. The analysis is conducted over a dataset obtained with a genetic algorithm-based optimization process for nominal WEC design. The results reveal a significant deviation in terms of robustness between the nominal Pareto set and those generated by setting different thresholds for robustness metrics, as well as between devices belonging to the same nominal Pareto frontier. This study elucidates the intrinsic need for incorporating robust optimization processes in WEC design.
... By applying Ideal and Anti-Ideal Compromise Programming, we can establish a logical mathematical relationship between the two contradictions [13,14]. Anti-ideal solutions represent the worst possible values for each objective individually, and the objective is to maximize the distance between all objectives and the anti-ideal values [15,16]. As a first contribution in this paper, we develop a two-criteria portfolio optimization model. ...
Portfolio optimization is an important topic in the financial management literature. This problem consists of managing wealth across available assets to gain maximum profit with the least amount of risk. In this paper, a novel approach for mean conditional value at risk (Mean-CVaR) as a bi-objective portfolio optimization problem is proposed. In order to address mean-CVaR, we utilized an ideal and anti-ideal compromise programming approach. Compared to compromise programming and goal programming, this approach is more efficient for solving the portfolio optimization problem, as it aims to increase the distance between solutions and anti-ideal criteria. This research utilizes a real-life case study from the Tehran Stock Exchange (TSE) to demonstrate the effectiveness of the proposed approach. The computational results determined the effectiveness of the solution methodology and it could be applied quite reliably in other engineering contexts without a significant degradation in performance.
... On the other hand, robust approaches for uncertain LCPs have been considered recently as well. The first rigorous analysis of robust LCPs can be found in [19,20], where the authors apply the concept of strict robustness [18] to LCPs, which has been used later in [16] in the context of Cournot-Bertrand equilibria in power networks. Moreover, in [13,14], LCPs have been studied using Γ-robustness as introduced in [3,4,17]; see [6,12] for some applications in power markets. ...
We consider adjustable robust linear complementarity problems and extend the results of Biefel et al. (SIAM J Optim 32:152–172, 2022) towards convex and compact uncertainty sets. Moreover, for the case of polyhedral uncertainty sets, we prove that computing an adjustable robust solution of a given linear complementarity problem is equivalent to solving a properly chosen mixed-integer linear feasibility problem.
... The concept of static RO was first proposed by Soyster in the 1970s, who studied a linear optimization problem with a box uncertainty set [42]. Later in the 1990s, static RO was formally introduced [7,8,18] and its computational advantage has resulted in its wide usage in applications, including in portfolio selection [26,47], scheduling [15], operations management [30], etc. ...
This paper explores a class of nonlinear Adjustable Robust Optimization (ARO) problems, containing here-and-now and wait-and-see variables, with uncertainty in the objective function and constraints. By applying Fenchel’s duality on the wait-and-see variables, we obtain an equivalent dual reformulation, which is a nonlinear static robust optimization problem. Using the dual formulation, we provide conditions under which the ARO problem is convex on the here-and-now decision. Furthermore, since the dual formulation contains a non-concave maximization on the uncertain parameter, we use perspective relaxation and an alternating method to handle the non-concavity. By employing the perspective relaxation, we obtain an upper bound, which we show is the same as the static relaxation of the considered problem. Moreover, invoking the alternating method, we design a new dual-based cutting plane algorithm that is able to find a reasonable lower bound for the optimal objective value of the considered nonlinear ARO model. In addition to sketching and establishing the theoretical features of the algorithms, including convergence analysis, by numerical experiments we reveal the abilities of our cutting plane algorithm in producing locally robust solutions with an acceptable optimality gap.
... As a result, we need to develop models immune to data uncertainty. Introduced by Soyster (1973), robust optimization has been one leading decision-making technique under uncertainty. It aims to provide a comprehensive solution that maintains close to the best performance limit concerning the change of the input data under all uncertainty scenarios. ...
The concept of the circular economy (CE) proposes eco-friendly principles for dealing with circularity problems in production systems and supply chains (SCs). To achieve circularity goals in SCs, CE principles assist decision-makers with reusing, remanufacturing, and recycling initiatives in the production process. However, previous meso- and macro-level circularity investigations reveal shortcomings in practical closed-loop solutions. This study aims to develop a closed-loop framework for assessing the circularity of sustainable SCs using network data envelopment analysis (NDEA). The closed-loop framework is created utilizing real-life indices of sustainable SCs, such as "recyclable undesirable outputs". To make the appraisal more realistic and enhance the validity and reliability of the results, the circularity results of SCs are assessed under data uncertainty. This allows SCs to be ranked based on their real circularity level, identifying sustainable and unsustainable SCs. The proposed framework offers decision-makers a practical evaluation tool to identify the highly circular SCs and establish circularity benchmarks for inefficient SCs. The proposed approach is scalable and applicable to real-world circularity evaluations in multistage SCs and production systems. The study concludes with a two-stage case study to show the practicality of the new framework.
... The key feature of robust optimization is that the latter realization is studied in some kind of worst-case paradigm. For more details we refer to the seminal paper by Soyster (1973) as well as to the books and surveys by Ben-Tal and Nemirovski (1998), Ben-Tal et al. (2009), Bertsimas et al. (2011), Bertsimas and den Hertog (2022), and Buchheim and Kurtz (2018). ...
Robust and bilevel optimization share the common feature that they involve a certain multilevel structure. Hence, although they model something rather different when used in practice, they seem to have a similar mathematical structure. In this paper, we analyze the connections between different types of robust problems (strictly robust problems with and without decision-dependence of their uncertainty sets, min-max-regret problems, and two-stage robust problems) as well as of bilevel problems (optimistic problems, pessimistic problems, and robust bilevel problems). It turns out that bilevel optimization seems to be more general in the sense that for most types of robust problems, one can find proper reformulations as bilevel problems but not necessarily the other way around. We hope that these results pave the way for a stronger connection between the two fields-in particular to use both theory and algorithms from one field in the other and vice versa.
... For a single objective uncertain optimization problem, strict robustness was first introduced in [9]. The robustness for single objective optimization problems has been extensively researched in [7,8,[10][11][12]. ...
In this paper, we find the flimsily robust weakly efficient solution to the uncertain vector optimization problem by means of the weighted sum scalarization method and strictly robust counterpart. In addition, we introduce a higher-order weak upper inner Studniarski epiderivative of set-valued maps, and obtain two properties of the new notion under the assumption of the star-shaped set. Finally, by applying the higher-order weak upper inner Studniarski epiderivative, we obtain a sufficient and necessary optimality condition of the vector-based robust weakly efficient solution to an uncertain vector optimization problem under the condition of the higher-order strictly generalized cone convexity. As applications, the corresponding optimality conditions of the robust (weakly) Pareto solutions are obtained by the current methods.
... The degree of conservatism of the solution is controlled by varying the protection parameters Γ TAC and Γ GWP , according to the reformulation proposed by Bertsimas and Sim [48]. The latter is an extension of the classic minmax formulation proposed by Soyster [49] and broadened by Ben-Tal et al. [50]; however, this linear formulation provides a flexible and less conservative solution by enforcing just a certain number of parameters to assume their worst cases simultaneously. The value of Γ may range from 0 to the maximum number of uncertain parameters |J|. ...
... First, according to Soyster's [49] robust model, we establish a robust model based on the box uncertainty set. Suppose a m is the mth row vector of the uncertainty parameter matrix A, where N m is the set of the uncertain part a mn in the mth row of matrix A. The uncertainty parameter a mn can vary in the interval [ā mn −â mn ,ā mn +â mn ], whereā mn is the nominal value,â mn is the perturbation value, andâ mn ≥ 0. Referring to the Soyster robust model, the uncertainty set of demand D i in constraint (4) can be expressed as follows: ...
The uncertainty of post-earthquake disaster situations can affect the efficiency of rescue site selection, material, and personnel dispatching, as well as the sustainability of related resources. It is crucial for decision-makers to make decisions to mitigate risks. This paper first presents a dual-objective model for locating emergency logistics facilities, taking into account location costs, human resource scheduling costs, transportation time, and uncertainties in demand and road conditions. Then, stochastic programming and robust optimization methods are utilized to cater to decision-makers with varying risk preferences. A risk-preference-based stochastic programming model is introduced to handle the potential risks of extreme disasters. Additionally, robust models are constructed for two uncertain environments. Finally, the study uses the Wenchuan earthquake as a case study for the pre-locating of emergency logistics facilities and innovatively compares the differences in the effects of models constructed using different uncertainty methods. Experimental results indicate that changes in weight coefficients and unit transportation costs significantly impact the objective function. This paper suggests that decision-makers should balance cost and rescue efficiency by choosing appropriate weight coefficients according to the rescue stage. It also shows that risk level and robust conservatism can significantly alter the objective function. While stochastic programming models offer economic advantages, robust optimization provides better robustness.
... In these environments, it is hard to specify detailed probability distributions for uncertain data. Soyster (1973) first developed robust optimization, but the solutions derived from this method were too conservative. To control the degree of conservatism, Bertsimas and Sim (2004) developed a new robust optimization method (called B&S method hereafter), in which only a subset of uncertain coefficients is in the worst scenarios. ...
This paper addresses a sequencing problem with uncertain task times in mixed model assembly lines. In this problem, task times are not known exactly but are given by intervals of their possible values. A mixed integer non-linear programming model is developed to minimize the utility work time, which is converted into a mixed integer linear programming (MILP) model to deal with small-sized instances optimally. Due to the NP-hardness of the problem, a discrete cuckoo search particle swarm optimization (DCSPSO) algorithm is developed. In the proposed algorithm, a particle position is updated by crossover and mutation operators in the discrete domain and discrete Levy flight is used to improve the solution quality further. Numerical experiments are conducted on the designed instances. The results indicate that the DCSPSO algorithm outperforms the exact method and the other three meta-heuristic algorithms. A case study of engine cylinder heads sequencing problem shows the proposed approach can obtain multiple solutions for decision-makers to choose according to the actual situation.
... Several common robust optimization frameworks are the worst-case hedge model (Soyster, 1973), minimax regret (minimizing the maximum regret), and using uncertainty sets [(ellipsoidal uncertainty sets (Ben-Tal & Nemirovski, 2002, 1999, or polyhedral uncertainty sets (Bertsimas & Sim, 2004]. The worst-case hedge model and the minimax regret model are the simplest to solve, but many times are overly pessimistic. ...
This article discusses a robust network interdiction problem considering uncertainties in arc capacities and resource consumption. The problem involves two players: an adversary seeking to maximize the flow of a commodity through the network and an interdictor whose objective is to minimize this flow. The interdictor plays first and selects network arcs to interdict, subject to a resource constraint. The problem is formulated as a bilevel problem, and an upper bound single level mix-integer linear formulation is derived. The upper bound formulation is solved using three heuristics tailored for this problem and the network structure, based on Lagrangian relaxation and Benders’ decomposition. On average, each heuristic provides a reduction in run time of at least 85% compared to a state-of-the-art solver. Enhanced Benders’ decomposition achieves a solution with an optimality gap of less than 5% for all tested instances. Sensitivity analyses are conducted for the level of uncertainty in network parameters and the uncertainty budget. Robust decisions are also compared to decisions not accounting for uncertainty to evaluate the value of robustness, showing a reduction in simulation maximum flows by as much as 89.5%.
Robust supply chain network design that considers supply resiliency, plays vital role in supply chain risk management in dealing with various operational and disruption risks. This study developed a novel three-stage decision approach to consider two echelons robust and resilient supply chain networks. We present a mixed-integer non-linear programming model with two objective functions. The objectives are maximization of SCN profit and maximization of resiliency, where robustness, agility, leanness, flexibility, and integrity can be defined as the five resiliency criteria. Fuzzy Simultaneous Evaluation of Criteria and Alternatives (FSECA) and Simple Multi-Attribute Rating technique (SMART) have been used to obtain the supplier resiliency and weighted importance of resilience criteria. Then, a robust optimization model is built based on uncertainty parameters considering supplier resiliency. A Non-dominated Sorting Genetic Algorithm (NSGAII) and Multi Objective Particle Swarm optimization (MOPSO) were used to solve the robust model on a large scale. parameters calibrated by the Taguchi method and five metrics of performance evaluation were considered to compare the meta-heuristic algorithms. We demonstrate the proposed NSGAII algorithm over a competing method based on five performance metrics. The research findings reveal the optimal level of robust supply chain networks based on algorithm performance and Taguchi analyses. Moreover, the results indicate that when profit increases, resilience can increase simultaneously.
The main goal of this paper is to present a new approach for measuring the performance of n-stage series systems in the presence of uncertain data, which are two challenging issues in evaluating the efficiency of Decision-Making Units (DMUs) using traditional Data Envelopment Analysis (DEA) models. By using a Network DEA model and its dual, as well as using Bertsimas et al.'s robustness technique, two Robust Network DEA have been presented. These models can display a range of DMUs performance with appropriate accuracy. Proposed models were used to determine the efficiency range of Iranian dairy companies' supply chain with three stages. The results show that the proposed models are applicable and effective. Total efficiency bounds are obtained with percentage deviations of 20%, 10% and 5%. The lower bounds have relative errors of 0.39, 0.23 and 0.12 and a correlation coefficient of more than 97%, and the upper bounds have relative error of 1.1, 0.84 and 0.62 and a correlation coefficient of about 90%. Therefore, the proposed model for calculating the lower bound is more accurate. The calculation of the efficiency bounds of the sub-stages also confirms this issue. Finally, the obtained results have been compared with the values obtained through a fuzzy three-stage DEA model, our results have a higher correlation coefficient and more accurate upper bounds.
An efficient municipal solid waste (MSW) system is critical to modern cities in order to enhance sustainability and liveability of urban life. With this aim, the planning phase of the MSW system should be carefully addressed by decision makers. However, planning success is dependent on many sources of uncertainty that can affect key parameters of the system, for example, the waste generation rate in an urban area. With this in mind, this article contributes with a robust optimization model to design the network of collection points (i.e. location and storage capacity), which are the first points of contact with the MSW system. A central feature of the model is a bi-objective function that aims at simultaneously minimizing the network costs of collection points and the required collection frequency to gather the accumulated waste (as a proxy of the collection cost). The value of the model is demonstrated by comparing its solutions with those obtained from its deterministic counterpart over a set of realistic instances considering different scenarios defined by different waste generation rates. The results show that the robust model finds competitive solutions in almost all cases investigated. An additional benefit of the model is that it allows the user to explore trade-offs between the two objectives.
This paper aims to present alternative characterizations for different types of set-valued robustness concepts. Equivalent scalar representations for various set order relations are derived when the sets are the union of sets. Utilizing these findings in conjunction with image space analysis, specific isolated sets are defined for different notions of robust solutions. These isolated sets serve as the basis for deriving both necessary and sufficient robust optimality conditions. The validity of the results is demonstrated through several illustrative examples. Additionally, the paper concludes with an application of our present approach to two-player zero-sum matrix games.
In this study, sufficient optimality criteria for a robust fractional interval-valued optimization problem (RP) is highlighted based on the idea of convexificators. The use of generalized invexity tools to support sufficient optimality conditions is demonstrated by an example. Also, saddle point condition of a Lagrange function is examined for RP. Furthermore, Mond–Weir-type robust dual model is constructed and by employing the concept of generalized invexity, usual duality results are discussed.
A robust optimization model is a paradigm for decision-making under uncertainty, where the parameters are given in the form of uncertainty sets. In this paper, we establish a robust optimization model for a captive repair shop. This paper is divided into two parts. In the first part, we present an uncertain linear optimization model involving the product of two uncertain parameters. Information associated with these uncertain parameters are modelled in the form of polyhedral and ellipsoidal uncertainty sets. The max-min regret approach is applied to solve the uncertain optimization model. In the second part, the robust optimization technique is applied to a multi-plant captive repair shop for repairing and overhauling multiple products over multiple periods. Numerical examples are presented to illustrate the theoretical results.
ResearchGate has not been able to resolve any references for this publication.