In practical applications of optimization it is common to have several conflicting objective functions to optimize. Frequently, these functions are subject to noise or can be of black-box type, preventing the use of derivative-based techniques.We propose a novel multiobjective derivative-free methodology, calling it direct multisearch (DMS), which does not aggregate any of the objective functions. Our framework is inspired by the search/poll paradigm of direct-search methods of directional type and uses the concept of Pareto dominance to maintain a list of nondominated points (from which the new iterates or poll centers are chosen). The aim of our method is to generate as many points in the Pareto front as possible from the polling procedure itself, while keeping the whole framework general enough to accommodate other disseminating strategies, in particular when using the (here also) optional search step. DMS generalizes to multiobjective optimization (MOO) all direct-search methods of directional type.We prove under the common assumptions used in direct search for single optimization that at least one limit point of the sequence of iterates generated by DMS lies in (a stationary form of) the Pareto front. However, extensive computational experience has shown that our methodology has an impressive capability of generating the whole Pareto front, even without using a search step.Two by-products of this paper are (i) the development of a collection of test problems for MOO and (ii) the extension of performance and data profiles to MOO, allowing a comparison of several solvers on a large set of test problems, in terms of their efficiency and robustness to determine Pareto fronts.

In order to correctly assess the biaxial fatigue material properties one must experimentally test different load conditions and stress levels. With the rise of new in-plane biaxial fatigue testing machines, using smaller and more efficient electrical motors, instead of the conventional hydraulic machines, it is necessary to reduce the specimen size and to ensure that the specimen geometry is appropriated for the load capacity installed. At the present time there are no standard specimen's geometries and the indications on literature how to design an efficient test specimen are insufficient. The main goal of this paper is to present the methodology on how to obtain an optimal cruciform specimen geometry, with thickness reduction in the gauge area, appropriated for fatigue crack initiation, as a function of the base material sheet thickness used to build the specimen. The geometry is optimized for maximum stress using several parameters, ensuring that in the gauge area the stress is uniform and maximum with two limit phase shift loading conditions. Therefore the fatigue damage will always initiate on the center of the specimen, avoiding failure outside this region. Using the Renard Series of preferred numbers for the base material sheet thickness as a reference, the reaming geometry parameters are optimized using a derivative-free methodology, called direct multi search (DMS) method. The final optimal geometry as a function of the base material sheet thickness is proposed, as a guide line for cruciform specimens design, and as a possible contribution for a future standard on in-plane biaxial fatigue tests.

No abstract

Locating and identifying points as global minimizers is, in general, a hard and time-consuming task. Difficulties increase in the impossibility of using the derivatives of the functions defining the problem. In this work, we propose a new class of methods suited for global derivative-free constrained optimization. Using direct search of directional type, the algorithm alternates between a search step, where potentially good regions are located, and a poll step where the previously located promising regions are explored. This exploitation is made through the launching of several instances of directional direct searches, one in each of the regions of interest. Differently from a simple multistart strategy, direct searches will merge when sufficiently close. The goal is to end with as many direct searches as the number of local minimizers, which would easily allow locating the global extreme value. We describe the algorithmic structure considered, present the corresponding convergence analysis and report numerical results, showing that the proposed method is competitive with currently commonly used global derivative-free optimization solvers.

No abstract

h i g h l i g h t sOperational strategy and sizing optimization of an integrated power and water supply system. Multiobjective optimization using the Direct Multisearch method. Significant increase in RES production of total electricity production is feasible. Significant cost and CO 2 emissions reductions are achievable. a b s t r a c tThe penetration of intermittent renewable energy sources, for instances wind power, in the power system of isolated islands is limited, even when there is large potential. The wind power that cannot be directly injected in the power grid is usually curtailed. In addition, some islands need to desalinate seawater to produce fresh water, increasing the pressure on the power system, because desalination needs electricity. Nevertheless, the water scarcity problem of an island can be part of the solution of the problem of its integration of intermittent renewable energy sources. To tackle this issue, a system was proposed to use the excess wind power in desalination units and in a pumped hydro storage, resulting in an integrated power and water supply system that would minimize the wind power curtailed. This paper proposes a methodology to optimize the size and operational strategy of this wind powered desalination and pumped hydro storage system. The objective is to minimize the total annualized production costs, maximize the percentage of renewable energy sources in the total power production and minimize the wind power curtailed. To solve this optimization problem, a derivative free multiobjective optimization method (Direct MultiSearch) is used. This methodology is applied to the integrated power and water supply system proposed for the island of S. Vicente, in Cape Verde. The results show that the penetration of renewable energy sources can reach 84% with a 27% decrease of power and water production costs and 67% decrease of CO 2 emissions, in relation to the values foreseen for 2020.

No abstract

In practical applications it is common to have several conflicting objective functions to optimize. Frequently, these functions are nondifferentiable or discontinuous, could be subject to numerical noise and/or be of black-box type, preventing the use of derivative-based techniques. In this paper we give an overview of some recent developments in Derivative-free Multiobjective Optimization. We introduce the basic concepts and ideas commonly considered for the algorithmic development in Multiobjective Optimization and review some recent classes of methods which do not make use of derivatives. In particular, we will focus on Direct Search Methods (DSM) of directional type and Evolutionary Multiobjective Optimization (EMO).