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In this paper, a numerical approximation method is developed to find approximate solutions to a class of constrained multi-objective optimization problems. All the functions of the problem are not necessarily convex functions. At each iteration of the method, a particular type of subproblem is solved using the trust region technique, and the step is evaluated using the notions of actual reduction and predicted reduction. A non-differentiable $ l_{\infty } $ l∞ penalty function restricts the constraint violations. An adaptive BFGS update formula is introduced. Global convergence of the proposed algorithm is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. Furthermore, it is justified that the proposed algorithm displays a super-linear convergence rate. Numerical results are provided to show the efficiency of the algorithm in the quality of the approximated Pareto front. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear scalarization function for a variable domination structure in an arbitrary linear space without assuming any particular topology. Conditions are provided under which the nonlinear scalarization function possesses several useful properties such as finiteness, properness, positive homogeneity, subadditivity, (strict) monotonicity, convexity or continuity. These properties are employed to characterize approximate efficiency in linear spaces. [ABSTRACT FROM AUTHOR]

In this paper, a globally convergent Newton-type proximal gradient method is developed for composite multi-objective optimization problems where each objective function can be represented as the sum of a smooth function and a nonsmooth function. The proposed method deals with unconstrained convex multi-objective optimization problems. This method is free from any kind of priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a subproblem is solved to find a suitable descent direction. The subproblem uses a quadratic approximation of each smooth function. An Armijo type line search is conducted to find a suitable step length. A sequence is generated using the descent direction and the step length. The global convergence of this method is justified under some mild assumptions. The proposed method is verified and compared with some existing methods using a set of test problems. [ABSTRACT FROM AUTHOR]

This paper deals with quasidifferentiable vector optimization problems involving invex functions wrt convex compact sets. We present vector variational-like inequalities of Minty type and of Stampacchia type in terms of quasidifferentials denoted by (QMVVLI) and (QSVVLI), respectively. By utilizing these variational inequalities, we infer vital and adequate optimality conditions for an efficient solution of the quasidifferentiable vector optimization problem involving invex functions wrt convex compact sets. We also establish various results for the solutions of the corresponding weak versions of the vector variational-like inequalities in terms of quasidifferentials. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the Continuous-Time Problem in an interval context and to obtain optimality conditions for this problem. In addition, we will find relationships between solutions of Interval Continuous-Time Problem and Interval Variational-like Inequality Problems, both Stampacchia and Minty type. Pseudo invex monotonicity condition ensures the existence of solutions of the problem. These results generalize similar conclusions obtained in Euclidean or Banach spaces inside classical mathematical programming problems or Continuous-Time Problems. We will finish generalizing the existence of Walrasarian equilibrium price model and the Wardrop's principle for traffic equilibrium problem to an environment of interval-valued functions. [ABSTRACT FROM AUTHOR]

Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years. Recently Delage et al. [Shortfall Risk Models When Information of Loss Function Is Incomplete, GERAD HEC, Montréal, 2018] consider a situation where a decision maker's true loss function in the definition of SR is unknown but it is possible to elicit a set of plausible utility functions with partial information and consequently propose a robust formulation of SR based on the worst-case utility function. In this paper, we extend this new stream of research to a multi-attribute prospect space since multi-attribute decision-making problems are ubiquitous in practical applications. Specifically, we introduce a preference robust multivariate utility-based shortfall risk measure (PRMSR) and demonstrate that it is law invariant and convex. We then apply the PRMSR to an optimal decision-making problem where the objective is to minimize the PRMSR of a vector-valued cost function and propose some numerical scheme for solving the resulting optimization problem in the case when the underlying exogenous uncertainty is finitely distributed. Finally, we discuss statistical robustness of the PRMSR based optimization model by examining qualitative stability of the estimator of the optimal value obtained with potentially contaminated data. A case study is carried out to examine the performance of the proposed robust model and numerical scheme. [ABSTRACT FROM AUTHOR]

In order to generate properly efficient solutions of a general multiobjective optimization problem, a unified direction approach was introduced by Ghaznavi et al. (Optim. Methods Softw., 2019). However, there are some deficiencies in this work, and a counterexample is given to show that properly efficient solutions of a general multiobjective optimization problem cannot be obtained via optimal solutions of the proposed scalarized problem. In the present paper, by adding slack variables and surplus variables, we modify the direction scheme proposed by Ghaznavi et al. (Optim. Methods Softw., 2019), and achieve necessary and sufficient conditions for (weakly and properly) efficient solutions of a general multiobjective optimization problem via optimal solutions of the corresponding scalarized problem. Further, the efficiency of our method is demonstrated by numerical examples. This work fills some gaps in the work of Ghaznavi et al. (Optim. Methods Softw., 2019). [ABSTRACT FROM AUTHOR]

The presented paper addresses scalarization techniques for multiobjective optimization problems. A new scalarization technique for solving a general multiobjective optimization problem is proposed. This scalarization method is called the unified direction method, because it generalizes the Pascoletti–Serafini approach (direction approach). Some theoretical results related to the suggested scalarization technique are established, especially the ability of the method for generating (properly, weakly) efficient solutions is shown. Moreover, the relation between approximate optimal solutions of the proposed single objective problem and approximate (weakly, properly) efficient solutions of the multiobjective optimization problem is investigated. In fact, easy-to-check, necessary and sufficient conditions for ϵ-(weak, proper) efficiency are obtained. Moreover, the effectiveness of the proposed approach is illustrated with some illustrative computational examples. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study how to solve a type of matrix game with interval payoffs. In this paper, the interval inequality relations and the concept of solutions of the matrix game with interval payoffs are defined. Based on the fuzzy ranking index defined, the solution of the matrix game with interval payoffs can be obtained through solving a pair of bi-objective linear programming models derived from the constructed auxiliary interval programming models. It is shown that the models proposed in this paper extend those of the classical matrix game. The validity and applicability of the proposed methodology are illustrated with a numerical example. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce the anchor point notion for multiobjective optimization problems. These points are special properly efficient points. We characterize anchor points via some theorems. The theoretical results lead to two tests which are able to recognize whether a given feasible point is an anchor point or not. One of these tests works by solving several linear programming problems and another one is done by solving a single mixed-integer programming problem. We compare the obtained procedures by implementing them on various randomly generated numerical examples. Some discussions about the density and connectedness of the set of nonanchor points are provided as well. [ABSTRACT FROM AUTHOR]

A Nonlinear Stepsize Control (NSC) framework has been proposed by Toint [Nonlinear stepsize control, trust regions and regularizations for unconstrained optimization, Optim.Methods Softw. 28 (2013), pp. 82–95] for unconstrained optimization, generalizing many trust-region and regularization algorithms. More recently, worst-case complexity bounds for the generic NSC framework were proved by Grapigliaet al.[On the convergence and worst-case complexity of trust-region and regularization methods for unconstrained optimization, Math. Program. 152 (2015), pp. 491–520] in the context of non-convex problems. In this paper, improved complexity bounds are obtained for convex and strongly convex objectives. [ABSTRACT FROM AUTHOR]

This paper first proposes a trust region algorithm to obtain a stationary point of unconstrained multiobjective optimisation problem. Under suitable assumptions, the global convergence of the new algorithm is established. We then extend the trust region method to solve the non-smooth multiobjective optimisation problem. [ABSTRACT FROM PUBLISHER]

In this paper, we give the vector versions of the concepts of approximate starshapedness, equi-subdifferentiability and pseudo-equi-subdifferentiability and establish relationships among approximate vector starshapedness, vector-equi-subdifferentiability and vector-pseudo-equi-subdifferentiability. We extend the concept ofε-quasi-efficient solutions in the context of multiobjective optimization problems involving approximately starshaped functions and use approximate vector variational inequalities of Stampacchia and Minty type in terms of Fréchet subdifferentials to characterize approximate efficient solutions. [ABSTRACT FROM AUTHOR]

Following the classical exponential penalty function method of mathematical programming, the exponential penalty function method of multiobjective programming problems (MOPP) is constructed and its convergence is proved. In addition, the approach is applied to solving a finite min-max MOPP. [ABSTRACT FROM AUTHOR]

In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarization finds the minimum 'distance' from a reference point, which is usually taken as a vertex of the current outer approximation, to the upper image through a given direction. We propose efficient methods to select the parameters (the reference point and direction vector) of the PS scalarization and analyse the effects of these on the overall performance of the algorithm. Different from the existing vertex selection rules from the literature, the proposed methods do not require solving additional single-objective optimization problems. Using some test problems, we conduct an extensive computational study where three different measures are set as the stopping criteria: the approximation error, the runtime, and the cardinality of the solution set. We observe that the proposed variants have satisfactory results, especially in terms of runtime compared to the existing variants from the literature. [ABSTRACT FROM AUTHOR]

In this manuscript, we consider multi-objective optimization problems with a cardinality constraint on the vector of decision variables and additional linear constraints. For this class of problems, we analyse necessary and sufficient conditions of Pareto optimality. We afterwards propose a Penalty Decomposition type algorithm, exploiting multi-objective descent methods, to tackle the aforementioned family of problems. We conduct a rigorous convergence analysis for the proposed method, where we prove that the produced sequence of points has limit points, each one being feasible and satisfying first-order optimality conditions. Numerical computational experiments, carried out on instances of relevant real-world problems such as sparse mean/variance portfolio selection and sparse regularized logistic regression, in their multi-objective formulation, show that the proposed procedure is effective at finding solutions forming good Pareto sets approximations. [ABSTRACT FROM AUTHOR]

We present an algorithm for approximately solving bounded convex vector optimization problems. The algorithm provides both an outer and an inner polyhedral approximation of the upper image. It is a modification of the primal algorithm presented by Löhne, Rudloff, and Ulus in 2014. There, vertices of an already known outer approximation are successively cutoff to improve the approximation error. We propose a new and efficient selection rule for deciding which vertex to cutoff. Numerical examples are provided which illustrate that this method may solve fewer scalar problems overall and therefore may be faster while achieving the same approximation quality. [ABSTRACT FROM AUTHOR]

We propose and analyse a nonmonotone quasi-Newton algorithm for unconstrained strongly convex multiobjective optimization. In our method, we allow for the decrease of a convex combination of recent function values. We establish the global convergence and local superlinear rate of convergence under reasonable assumptions. We implement our scheme in the context of BFGS quasi-Newton method for solving unconstrained multiobjective optimization problems. Our numerical results show that the nonmonotone quasi-Newton algorithm uses fewer function evaluations than the monotone quasi-Newton algorithm. [ABSTRACT FROM AUTHOR]

The constrained vector optimization problem minC f (x), g(x) ∈ −K, h(x) = 0 is studied, where f : Rn → Rm, g : Rn → Rp and h : Rn → Rq are C1,1 functions, and C ⊂ Rm and K ⊂ Rp are closed convex cones. In terms of the second-order Dini set-valued directional derivative, second-order necessary conditions a point x0 to be a weakly efficient point and second-order sufficient conditions x0 to be an isolated minimizer of order two are established. [ABSTRACT FROM AUTHOR]