Scalarization Technique Research Articles

A comparative study of different algorithms and scalarization techniques for constructing the Pareto front of multiobjective optimization problems

This paper presents an analysis and comparison of various algorithms applied to different scalarization methods for multiobjective optimization problems (MOPs). At first, some theoretical results are provided on the relation between (weakly) efficient solutions of an MOP and optimal solutions of the related numerical method. Moreover, we consider some deficiencies in different scalarization approaches given in the literature and try to fill some gaps in these works. Hence, by considering appropriate limitations, we provide sufficient conditions for (ε-)properly efficient and (ε-)efficient solutions of an MOP via scalarization techniques. Then, an algorithm for approximating the Pareto front of MOPs is presented. The main purpose of this algorithm is to generate efficient points on the Pareto front with a uniform distribution. One advantage of this algorithm, compared to some other algorithms, is that in each iteration of the algorithm, more than one efficient point located on the Pareto border can be generated. The new algorithm is implemented by applying the modified Pascoletti–Serafini, the unified direction, and the modified normal boundary intersection scalarization techniques, and then the procedures are considered on different test problems including (non-)convex and discrete Pareto fronts. The capability of the numerical method is shown by comparing the results with the algorithms in the literature. To compare the algorithms, measures of coverage and spacing metric indicators are used.

Multi-objective optimization for fast charging design of lithium-ion batteries using constrained Bayesian optimization

Fast charging of lithium-ion battery accounting for both charging time and battery degradation is key to modern electric vehicles. The challenges of fast charging optimization are (i) the high dimensionality of the space of possible charging protocols while the experiment budget is often limited; and (ii) the limited quantitative description of battery capacity fade mechanisms. This article proposes a data-driven multi-objective charging approach to minimize charging time while maximizing battery cycle life, in which a Chebyshev scalarization technique is used to transform the multi-objective optimization problem into a group of single objective problems, and a constrained Bayesian optimization (BO) is then utilized to effectively explore the parameter space of charging current as well as handle the constraint of charging voltage. Moreover, continuous-varied-current charging protocols are introduced into the proposed charging optimization approach by the utilization of polynomial expansion technique. The effectiveness of the proposed charging approach is demonstrated on a porous electrode theory-based battery simulator. The results show that the proposed constrained BO-based approach possesses superior charging performance and higher sample efficiency, compared with the state-of-the-art baselines including constrained optimization by linear approximations (COBYLA) and covariance matrix adaptation evolutionary strategy (CMA-ES). In addition, the increase in the charging performance and its uncertainty with an increasing number of degrees of freedom used in charging protocols is discussed.

Necessary optimality conditions for a semivectorial bilevel optimization problem via a revisited Charnes–Cooper scalarization

In this paper, we investigate a multiobjective bilevel optimization problem with a vector-valued lower-level objective function. We revisit the Charnes–Cooper scalarization technique for multi-objective programs and use optimal value reformulation to transform the scalar problem into a one-level optimization problem. Using Mordukhovich's generalized differentiation calculus, a new subdifferential estimate and Lipschitz condition for the optimal value function of this problem are developed as a result of this reformulation. In addition, we construct first-order necessary optimality conditions in the smooth setting.

Ordinal optimization through multi-objective reformulation

We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible solutions. We review different optimality concepts for ordinal optimization problems and discuss their similarities and differences. We then focus on two prevalent optimality concepts that are shown to be equivalent. Our main focus lies on the investigation of a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost coefficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman’s principle of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We investigate the interrelation between scalarization techniques and methods based on the hypervolume indicator when applied to the ordinal and the transformed problem, respectively. Furthermore, we extend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.

Multiobjective Optimization for Adaptive Offloading in Distributed Multiuser MIMO Cell-Free 6G Networks

By offloading some computational tasks to the edge server, edge computing can help relieve the increasing computation burden of mobile users and improve the quality of experience of users. In this paper, we investigate the computation offloading in edge computing-enabled multiuser cell-free (CF) multi-input multi-output (MIMO) networks with multiple building baseband units (BBUs). The user association, transmit covariances, CPU-cycle frequencies, and allocated bandwidths are jointly optimized to minimize two objectives: the energy consumption of mobile devices (MDs) and execution delay of tasks. We formulate it as a vector optimization problem and adopt the scalarization technique to transform it into a scalar mixed integer nonlinear programming (MINLP). Then, to solve the MINLP, we introduce user sorting into the framework of branch and bound and propose a sorted MD association (MDA) method. Two key issues are tackled in the proposed MDA, i.e., the lower bound of the MINLP and the updation of incumbent solution. For the first issue, we present a Lagrangian dual relaxation algorithm based on subgradient projection, while the second one involving another nonconvex minimization problem, we propose a successive convex approximation (SCA) based algorithm to solve it. Finally, extensive simulations demonstrate that the proposed method is able to reduce the system cost significantly and achieve better system performance by comparing with the benchmark schemes.

A Generalized Multiple Criteria Data-Fitting Model With Sparsity and Entropy With Application to Growth Forecasting

In this article, we present an extended data-fitting model which involves different and conflicting criteria, and we propose an algorithm based on a scalarization technique to solve it. Our model integrates in a unique framework three different criteria, namely, a data-fitting term, and the entropy and the sparsity of the set of unknown parameters. This model can be analyzed by means of multiple criteria decision-making techniques. We then validate the proposed modified algorithm using two computational experiments: We analyze the problem of handwritten digit recognition using a logistic regression model and a deep neural network model, respectively. In the final part of the article, we employ this methodology to forecasting instead. Given the importance of forecasting techniques to predict the future, which in turn can lead to positive impacts on firm performance, we propose two numerical experiments focusing on the forecast of the US GDP. In the first one, we proceed by means of a modified iterated function system with grayscale maps-type fractal operator, and, in the second one, we implement a modified neural network-based model.

On the study of multistage stochastic vector quasi-variational problems

This paper focuses on the study of multistage stochastic vector generalized quasi-variational inequalities with a variable ordering structure. The proposed multistage stochastic vector quasi-variational problems are defined in a suitable functional setting relative to a finite set of final possible states and certain information fields; these formulations are a multicriteria extension of the multistage stochastic variational inequalities. A relevant aspect of these problems is the presence of the nonanticipativity constraints on the variables of the problem; stage by stage, these constraints impose the measurability with respect to the information field at that stage. Without requiring any assumption of monotonicity, we prove some existence results by using a nonlinear scalarization technique. On this basis, we analyze multistage stochastic vector Nash equilibrium problems: as an example, we focus on a suitable multistage stochastic bicriteria Cournot oligopolistic model.

Taming Lagrangian chaos with multi-objective reinforcement learning.

We consider the problem of two active particles in 2D complex flows with the multi-objective goals of minimizing both the dispersion rate and the control activation cost of the pair. We approach the problem by means of multi-objective reinforcement learning (MORL), combining scalarization techniques together with a Q-learning algorithm, for Lagrangian drifters that have variable swimming velocity. We show that MORL is able to find a set of trade-off solutions forming an optimal Pareto frontier. As a benchmark, we show that a set of heuristic strategies are dominated by the MORL solutions. We consider the situation in which the agents cannot update their control variables continuously, but only after a discrete (decision) time, [Formula: see text]. We show that there is a range of decision times, in between the Lyapunov time and the continuous updating limit, where reinforcement learning finds strategies that significantly improve over heuristics. In particular, we discuss how large decision times require enhanced knowledge of the flow, whereas for smaller [Formula: see text] all a priori heuristic strategies become Pareto optimal.

Multi-objective optimization for integrated sugarcane cultivation and harvesting planning

Sugarcane and its by-products make a relevant contribution to the world economy. In particular, the sugar-energy industry is affected by the timing of sugarcane cultivation and harvesting from which sucrose and bio-energy are produced. We address this issue by proposing a mixed-integer non-linear programming model to schedule planting and harvesting operations for different varieties of sugarcane. The decisions to be made include the choice of sugarcane varieties to be grown on a given set of plots, the periods for their cultivation, the subsequent harvesting periods, and the type of harvesting equipment. These decisions are subject to various constraints related to matching cultivation periods with harvesting periods according to the maturity cycles of the selected sugarcane varieties, the availability of harvesting machinery, the demand for sucrose and fiber, and further technical requirements. The tactical cultivation and harvesting plans to be determined account for three conflicting objectives, namely maximization of the total sucrose and fiber production, minimization of the total time devoted to harvesting, and minimization of the total cost of transporting the harvesting equipment. We develop a tailored exact method based on the augmented Chebyshev scalarization technique extended with a mechanism for identifying an initial feasible integer solution that greatly helps reduce the computational effort for obtaining Pareto-optimal solutions. Our computational study with instances that reflect the current cultivation and harvesting practices in Brazil demonstrate the effectiveness of the proposed methodology. In addition, a comparative analysis reveals the trade-offs achieved by alternative planting and harvesting schedules, thereby facilitating the decision-making process.

Efficient scalarization in multiobjective optimal control of a nonsmooth PDE

This work deals with the efficient numerical characterization of Pareto stationary fronts for multiobjective optimal control problems with a moderate number of cost functionals and a mildly nonsmooth, elliptic, semilinear PDE-constraint. When “ample” controls are considered, strong stationarity conditions that can be used to numerically characterize the Pareto stationary fronts are known for our problem. We show that for finite dimensional controls, a sufficient adjoint-based stationarity system remains obtainable. It turns out that these stationarity conditions remain useful when numerically characterizing the fronts, because they correspond to strong stationarity systems for problems obtained by application of weighted-sum and reference point techniques to the multiobjective problem. We compare the performance of both scalarization techniques using quantifiable measures for the approximation quality. The subproblems of either method are solved with a line-search globalized pseudo-semismooth Newton method that appears to remove the degenerate behavior of the local version of the method employed previously. We apply a matrix-free, iterative approach to deal with the memory and complexity requirements when solving the subproblems of the reference point method and compare several preconditioning approaches.

On the exactness of the ε-constraint method for biobjective nonlinear integer programming

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.

Robust duality for nonconvex uncertain vector optimization via a general scalarization

ABSTRACT This paper concentrates on robust duality relations for uncertain cone-constrained vector optimization problems in more general nonconvex settings. First, different from the existing results, a new class of generalized Lagrange functions of the considered problem is introduced by combining the image space analysis method and scalarization technique. Then, the Lagrange robust vector dual problem is formulated. Subsequently, the results of robust weak duality, strong duality and converse duality are given respectively, which characterize vector dual relations between the primal worst and dual best problems. Simultaneously, some examples are given to illustrate our results.

Bi-objective optimization for road vertical alignment design

This manuscript defines a bi-objective optimization model to finds road profiles that optimize the road construction cost and the vehicle operating costs, specifically the fuel consumption. The research implements and validates the formula for the fuel consumption cost. It further presents and examines a variety of well-known methods: three classical scalarization techniques (the ɛ-constraint method, weighted sum method, and weighted metric methods) and two evolutionary methods (NSGA-II and FP-NSGA-II). Moreover, to accelerate the performance of the chosen scalarization approaches, a warm start strategy is proposed.Numerical experiments are performed on 30 road samples for Caterpillar 793D off-highway trucks to determine the most robust approach for the proposed problem. The results are analyzed using the commonly-used performance indicators: hypervolume (to assess the convergence of solutions), spacing (to assess the diversity of solutions), and CPU time (to assess the speed). The research finds that the warm start strategy improves the performance of all the scalarization techniques and concludes that the most promising method for the proposed problem is the ɛ-constraint method with a warm start.

Technical note on the existence of solutions for generalized symmetric set-valued quasi-equilibrium problems utilizing improvement set

In this paper, we establish some existence results for the solution of the generalized symmetric set-valued quasi-equilibrium problem (GSSQEP). The new forms of GSSQEP via improvement set and scalar generalized symmetric set-valued quasi-equilibrium problem (GSSQEP for short) are also introduced. By using Kakutani–Fan–Glicksberg fixed point method, maximal element principle and nonlinear scalarization technique, we develop two classes of sufficient conditions for the existence of solutions to GSSQEP. The drawback of some existing work for this problem is overcome. Moreover, some applications to the symmetric set-valued equilibrium problem (SSEP), symmetric vector quasi-equilibrium problem (SVQEP) and the set-valued equilibrium problem (SEP) are also given in this paper. Our results improve a few existing ones in Fakhar and Zafarani [Generalized symmetric vector quasiequilibrium problems. J Optim Theory Appl. 2008;136:397–409], Farajzadeh et al. [On the existence of solutions of symmetric vector equilibrium problems via nonlinear scalarization. Bull Iran Math Soc. 2019;45:35–58], Fu [Symmetric vector quasi-equilibrium problems. J Math Anal Appl. 2003;285:708–713], Han et al. [Existence of solutions for symmetric vector set-valued quasi-equilibrium problems with applications. Pacific J Optim. 2018;14:31–49].

Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

In this paper, we consider, a new nonlinear scalarization function in vector spaces which is a generalization of the oriented distance function. Using the algebraic type of closure, which is called vector closure, we introduce the algebraic boundary of a set, without assuming any topology, in our context. Furthermore, some properties of this algebraic boundary set are given and present the concept of the oriented distance function via this set in the concept of vector optimization. We further investigate Q-proper efficiency in a real vector space, where Q is some nonempty (not necessarily convex) set. The necessary and sufficient conditions for Q-proper efficient solutions of nonconvex optimization problems are obtained via the scalarization technique. The scalarization technique relies on the use of two different scalarization functions, the oriented distance function and nonconvex separation function, which allow us to characterize the Q-proper efficiency in vector optimization with and without constraints.

Multi-objective models for the forest harvest scheduling problem in a continuous-time framework

In this study we present several multi-objective models for forest harvest scheduling in forest with single-species, even-aged stands using a continuous formulation. We seek to maximize economic profitability and even-flow of timber harvest volume, both for the first rotation and for the regulated forest. For that, we design new metrics that allow working with continuous decision variables, namely, the harvest time of each stand. Unlike traditional combinatorial formulations, this avoids dividing the planning horizon into periods and simulating alternative management prescriptions before the optimization process. We propose to combine a scalarization technique (weighting method) with a gradient-type algorithm (L-BFGS-B) to obtain the Pareto frontier of the problem, which graphically shows the relationships (trade-offs) between objectives, and helps the decision makers to choose a suitable weighting for each objective. We compare this approach with the widely used in forestry multi-objective evolutionary algorithm NSGA-II. We analyze the model in a Eucalyptus globulus Labill. forest of Galicia (NW Spain). The continuous formulation proves robust in forests with different structures and provides better results than the traditional combinatorial approach. For problem solving, our proposal shows a clear advantage over the evolutionary algorithm in terms of computational time (efficiency), being of the order of 65 times faster for both continuous and discrete formulations.

Solving Equilibrium Problems with Bifunctions Having the Property of Negative Transitivity

In this article, we introduce the notion of bifunction with the property of negative transitivity and, using the Berge-Klee intersection theorem, we establish existence theorems of the solutions for the equilibrium problems, when the involved bifunction has this property. Then, using standard scalarization techniques, we obtain existence criteria of the solutions for vector equilibrium problems. The last section of the article is devoted to the scalar and vector quasi-equilibrium problems.

Optimality and duality for complex multi-objective programming

<p style='text-indent:20px;'>We consider a complex multi-objective programming problem (CMP). In order to establish the optimality conditions of problem (CMP), we introduce several properties of optimal efficient solutions and scalarization techniques. Furthermore, a certain parametric dual model is discussed, and their duality theorems are proved.</p>

Algorithms for generating Pareto fronts of multi-objective integer and mixed-integer programming problems

Multi-objective integer or mixed-integer programming problems typically have disconnected feasible domains, making the task of constructing an approximation of the Pareto front challenging. The present article shows that certain algorithms that were originally devised for continuous problems can be successfully adapted to approximate the Pareto front for integer, and mixed-integer, multi-objective problems. Relationships amongst various scalarization techniques are established to motivate the choice of a particular scalarization in these algorithms. The proposed algorithms are tested by means of two-, three- and four-objective integer and mixed-integer problems, and comparisons are made. In particular, a new four-objective algorithm is used to solve a rocket injector design problem with a discrete variable, which is a challenging mixed-integer programming problem.

An Analytical Study in Multi-physics and Multi-criteria Shape Optimization

A simple multi-physical system for the potential flow of a fluid through a shroud, in which a mechanical component, like a turbine vane, is placed, is modeled mathematically. We then consider a multi-criteria shape optimization problem, where the shape of the component is allowed to vary under a certain set of second-order Hölder continuous differentiable transformations of a baseline shape with boundary of the same continuity class. As objective functions, we consider a simple loss model for the fluid dynamical efficiency and the probability of failure of the component due to repeated application of loads that stem from the fluid’s static pressure. For this multi-physical system, it is shown that, under certain conditions, the Pareto front is maximal in the sense that the Pareto front of the feasible set coincides with the Pareto front of its closure. We also show that the set of all optimal forms with respect to scalarization techniques deforms continuously (in the Hausdorff metric) with respect to preference parameters.