The classical Sard theorem states that the set of critical values of a $C^{k}$-map from an open set of $\R^n$ to $\R^p$ ($n\geq p$) has Lebesgue measure zero provided $k\geq n-p+1$. In the recent paper by Barbet, Dambrine, Daniilidis and Rifford, the so called “preparatory Sard theorem” for a compact countable set $I$ of $C^k$ … Read more
In this paper, a new method for determining all minimal representations of a face of a polyhedron is proposed. A main difficulty for determining prime and minimal representations of a face is that the deletion of one redundant constraint can change the redundancy of other constraints. To reduce computational efforts in finding all minimal representations … Read more
In real life applications optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one … Read more
The extension of the Hager-Zhang (HZ) nonlinear conjugate gradient method for vector optimization is discussed in the present research. In the scalar minimization case, this method generates descent directions whenever, for example, the line search satisfies the standard Wolfe conditions. We first show that, in general, the direct extension of the HZ method for vector … Read more
In recent years, the development of new algorithms for multiobjective optimization has considerably grown. A large number of performance indicators has been introduced to measure the quality of Pareto front approximations produced by these algorithms. In this work, we propose a review of a total of 63 performance indicators partitioned into four groups according to … Read more
There has been numerous amount of studies on proper Pareto points in multiobjective optimization theory. Geoffrion proper points are one of the most prevalent form of proper optimality. Due to some convergence issues a restricted version of these proper points, Geoffrion proper points with preset bounds has been introduced recently. Since solution of any algorithm … Read more
In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the … Read more
A set of 78 test examples is presented for the trust region method MHT described in J. Thomann, G. Eichfelder, A trust region algorithm for heterogeneous multi-objective optimization, 2018 (available as preprint: http://optimization-online.org/DB_HTML/2018/03/6495.html) . It is designed for multi-objective heterogeneous optimization problems where one of the objective functions is an expensive black-box function, for example … Read more
Consider the multi-objective ranking and selection (MORS) problem in which we select the Pareto-optimal set from a finite set of systems evaluated on three or more stochastic objectives. Solving this problem is difficult because we must determine how to allocate a simulation budget among the systems to minimize the probability that any systems are misclassified. … Read more
In a recent paper, Lucambio Pérez and Prudente extended the Wolfe conditions for the vector-valued optimization. Here, we propose a line search algorithm for finding a step-size satisfying the strong Wolfe conditions in the vector optimization setting. Well definiteness and finite termination results are provided. We discuss practical aspects related to the algorithm and present … Read more