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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 objective 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. [PUBLICATION ABSTRACT]

This paper addresses the solution of bound-constrained optimization problems using algorithms that require only the availability of objective function values but no derivative information. We refer to these algorithms as derivative-free algorithms. Fueled by a growing number of applications in science and engineering, the development of derivative-free optimization algorithms has long been studied, and it has found renewed interest in recent time. Along with many derivative-free algorithms, many software implementations have also appeared. The paper presents a review of derivative-free algorithms, followed by a systematic comparison of 22 related implementations using a test set of 502 problems. The test bed includes convex and nonconvex problems, smooth as well as nonsmooth problems. The algorithms were tested under the same conditions and ranked under several criteria, including their ability to find near-global solutions for nonconvex problems, improve a given starting point, and refine a near-optimal solution. A total of 112,448 problem instances were solved. We find that the ability of all these solvers to obtain good solutions diminishes with increasing problem size. For the problems used in this study, TOMLAB/MULTIMIN, TOMLAB/GLCCLUSTER, MCS and TOMLAB/LGO are better, on average, than other derivative-free solvers in terms of solution quality within 2,500 function evaluations. These global solvers outperform local solvers even for convex problems. Finally, TOMLAB/OQNLP, NEWUOA, and TOMLAB/MULTIMIN show superior performance in terms of refining a near-optimal solution.

With the soaring demand for electric power and the limited spinning reserve in the power system in Taiwan, the comprehensive management of both thermal power generation and load demand turns out to be a key to achieving the robustness and sustainability of the power system. This paper aims to design a demand bidding (DB) mechanism to collaborate between customers and suppliers on demand response (DR) to prevent the risks of energy shortage and realize energy conservation. The concurrent integration of the energy, transmission, and reserve capacity markets necessitates a new formulation for determining schedules and marginal prices, which is expected to enhance economic efficiency and reduce transaction costs. To dispatch energy and reserve markets concurrently, a hybrid approach of combining dynamic queuing dispatch (DQD) with direct search method (DSM) is developed to solve the extended economic dispatch (ED) problem. The effectiveness of the proposed approach is validated through three case studies of varying system scales. The impacts of tie-line congestion and area spinning reserve are fully reflected in the area marginal price, thereby facilitating the determination of optimal load reduction and spinning reserve allocation for demand-side management units. The results demonstrated that the multi-area bidding platform proposed in this paper can be used to address issues of congestion between areas, thus improving the economic efficiency and reliability of the day-ahead market system operation. Consequently, this research can serve as a valuable reference for the design of the demand bidding mechanism.

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function  f : R → R which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points  ( x k ) k ∈ N converging to a point  x ∗ where  f is discontinuous, lower semicontinuous and whose objective function value  f ( x ∗ ) is strictly less than  lim k → ∞ f ( x k ) . Moreover the dDSM generates trial points in only one of the continuity sets of  f near  x ∗ . This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample.

This work introduces Inter-DS , a blackbox optimization algorithmic framework for computationally expensive constrained multi-fidelity problems. When applying a direct search method to such problems, the scarcity of feasible points may lead to numerous costly evaluations spent on infeasible points. Our proposed algorithm addresses this issue by leveraging multi-fidelity information, allowing for premature interruption of an evaluation when a point is estimated to be infeasible. These estimations are controlled by a biadjacency matrix, for which we propose a construction. The proposed method acts as an intermediary component bridging any non multi-fidelity direct search solver and a multi-fidelity blackbox problem, giving the user freedom of choice for the solver. A series of computational tests are conducted to validate the approach. The results show a significant improvement in solution quality when an initial feasible starting point is provided. When this condition is not met, the outcomes are contingent upon specific properties of the blackbox.

Cancer therapeutic vaccines are used to strengthen a patient’s own immune system by amplifying existing immune responses. Intralesional administration of the bacteria-based emm55 vaccine together with the PD1 checkpoint inhibitor produced a strong anti-tumor effect against the B16 melanoma murine model. However, it is not trivial to design an optimal order and frequency of injections for combination therapies. Here, we developed a coupled ordinary differential equations model calibrated to experimental data and used the mesh adaptive direct search method to optimize the treatment protocols of the emm55 vaccine and anti-PD1 combined therapy. This method determined that early consecutive vaccine injections combined with distributed anti-PD1 injections of decreasing separation time yielded the best tumor size reduction. The optimized protocols led to a twofold decrease in tumor area for the vaccine-alone treatment, and a fourfold decrease for the combined therapy. Our results reveal the tumor subpopulation dynamics in the optimal treatment condition, defining the path for efficacious treatment design. Similar computational frameworks can be applied to other tumors and other combination therapies to generate experimentally testable hypotheses in a fairly unrestricted and inexpensive setting.

A derivative-free optimization (DFO) method is an optimization method that does not make use of derivative information in order to find the optimal solution. It is advantageous for solving real-world problems in which the only information available about the objective function is the output for a specific input. In this paper, we develop the framework for a DFO method called the DQL method. It is designed to be a versatile hybrid method capable of performing direct search, quadratic-model search, and line search all in the same method. We develop and test a series of different strategies within this framework. The benchmark results indicate that each of these strategies has distinct advantages and that there is no clear winner in the overall performance among efficiency and robustness. We develop the Smart DQL method by allowing the method to determine the optimal search strategies in various circumstances. The Smart DQL method is applied to a problem of solid-tank design for 3D radiation dosimetry provided by the UBCO (University of British Columbia—Okanagan) 3D Radiation Dosimetry Research Group. Given the limited evaluation budget, the Smart DQL method produces high-quality solutions.

To improve the crashworthiness of vehicles, the crashworthiness of the vehicle structure itself has to be optimised. Through the collision analysis of a certain high-speed train, this research found that the front-end structure is most important in the crashworthiness optimisation design of the vehicle; the constitutive material models required for this numerical simulation of an entire vehicle were obtained by performing loading tests at different strain rates; according to the highly non-linear characteristics of the ensuing structural deformation under impact loading, this research used specific energy absorption (SEA) as an objective function to construct a multi-parameter optimisation model of the front-end structure of the vehicle . Based on this, the optimisation analysis was conducted. In the optimisation, the optimal SEA value (3.6988 kJ/kg) of the structure is obtained by 130-step iteration using a modified method of feasible directions (MMFD)—a gradient optimisation method; the optimal value obtained after 101 iterations by applying a direct search method—Hooke-Jeeves (HJ) algorithm is 3.6454 kJ/kg; and the optimal value acquired after 192 iterations of a global optimisation method—adaptive simulated annealing (ASA)—is 3.6132 kJ/kg. Moreover, the optimum results were validated by collision analysis of the optimal structure using a MMFD model. The variation analysis of the structural SEA with each variable show that the optimisation model is able to extend the range of each design variable.

Direct search methods are mainly designed for use in problems with no equality constraints. However, there are many instances where the feasible set is of measure zero in the ambient space and no mesh point lies within it. There are methods for working with feasible sets that are (Riemannian) manifolds, but not all manifolds are created equal. In particular, reductive homogeneous spaces seem to be the most general space that can be conveniently optimized over. The reason is that a ‘law of motion’ over the feasible region is also given. Examples include Rn and its linear subspaces, Lie groups, and coset manifolds such as Grassmannians and Stiefel manifolds. These are important arenas for optimization, for example, in the areas of image processing and data mining. We demonstrate optimization procedures over general reductive homogeneous spaces utilizing maps from the tangent space to the manifold. A concrete implementation of the probabilistic descent direct search method is shown. This is then extended to a procedure that works solely with the manifold elements, eliminating the need for the use of the tangent space.

This paper presents a class of nonmonotone Direct Search Methods that converge to stationary points of unconstrained and boxed constrained mixed-integer optimization problems. A new concept is introduced: the quasi-descent direction. A point x is stationary on a set of search directions if there exists no feasible qdd on that set. The method does not require the computation of derivatives nor the explicit manipulation of asymptotically dense matrices. Preliminary numerical experiments carried out on small to medium problems are encouraging.