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Abstract The aim of this paper is to solve optimal design problems for industrialapplications when the objective function value requires the evaluation of expensivesimulation codes and its first derivatives are not available. In order to achieve this goalwe propose two new algorithms that draw inspiration from two existing approaches:a filled function based algorithm and a Particle Swarm Optimization method. In orderto test the efficiency of the two proposed algorithms, we perform a numerical compar-ison both with the methods we drew inspiration from, and with some standard GlobalOptimization algorithms that are currently adopted in industrial design optimization.Finally, a realistic ship design problem, namely the reduction of the amplitude of theheave motion of a ship advancing in head seas (a problem connected to both safetyand comfort), is solved using the new codes and other global and local derivative-chosen filled function based algorithms because they have interesting theoreticalproperties (see Lucidi and Piccialli 2002;Xuetal. 2001), even though they are lessused in practice. In the new algorithm DDFPSO we introduce a local search phasebased on the derivative free minimization method (DF) described in Lucidi and Scian-drone (2002).59522
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As concerns FILLDIR we introduce a new filled function and weanalyze its theoretical properties. Furthermore we propose a global optimization al-gorithm based on this new filled function, that roughly speaking is a deterministicmulti-start approach. In particular, the filled function and the original function areminimized by means of the derivative free minimization method (DF) described inLucidi and Sciandrone (2002).The new solvers are compared with a standard PSO method, with the filled func-tion based algorithm proposed in Lucidi and Piccialli (2002), and also against stan-dard methods which are currently used in industrial optimal design area on a set ofwell known test problems. The standard GO algorithms are chosen from differentclasses of GO methods, i.e., decomposition methods, genetic algorithms, clusteringtechniques and multi-start simulated annealing algorithms.Finally, we compare the enhanced GO algorithms with others global and local(DF described in Lucidi and Sciandrone (2002) and NEWUOA described in Powell(2004)) optimization methods on a real-world ship design problem, which consistsin minimizing the vertical motion of the center of gravity (heave motion) of a shipadvancing at fixed speed in head seas.We conclude this section by formally stating the constrained GO problem thatarises from the ship design problem:min f(x)g(x) ≤ 0,l ≤ x ≤ u,(1)where x,l,u ∈ Rn, f : Rn →R and g : Rn →Rm are twice continuously differen-tiable functions, even though we assume that the derivatives are not available.2 A new filled function approachIn this section we propose a new filled function algorithm to solve problem (1). Thisproblem has a twofold difficulty. On the one hand, the objective function presentsmultiple local minima and among them the global minimum point is sought. On theother hand, the presence of nonlinear constraints substantially increases the difficultyof the problem by introducing a further requirement on the solution points. As a firststep toward the definition of an algorithm based on a filled function, we initiallyignore all the constraints g(x) ≤ 0 and l ≤ x ≤ u, thus we refer to the problemminx∈Rnf(x). (2)We require the following assumption to hold true in this section.
Assumption 1 For any α ∈ R, the level set Lf (α) ={x ∈ Rn : f(x) ≤ α} is compact. This is a minimal requirement that is in general satisfied by any optimal designproblem.The main idea of an algorithm based on a filled function is that, once deter-mined a stationary point x∗1 of problem (2), a perturbation of the objective func-tion, called filled function, is built such that its stationary points satisfy the relationf(x)<f(x∗1 ). In this way, by minimizing the filled function it is possible to find astationary point of the filled function with a value of the original objective functionlower than f(x∗1 ). More formally, a scheme of a general algorithm based on a filledfunction can be stated as follows:Algorithm FILLEDData: x0 ∈ Rn. Set k←0.Step 1. Compute x∗k by applying a local minimization algorithm to the solution ofproblem (2) starting from xk .Step 2. Define a filled function Q(x, x∗k ).Step 3. Choose a point ¯ x and findy ∈ arg minx∈RnQ(x, x∗k )by using a local minimization algorithm starting from ¯ x.Step 4. If f(y)<f(x∗k ) then set xk+1 ←y, k←k +1 and goto Step 1.Otherwise goto Step 3.End AlgorithmWe do not specify a stopping criterion, since it can vary in practice from the ap-plication. For example, it can depend on the maximum allowed number of functionevaluations, on the CPU time or on the numbers of failures at Step 4.In the following we introduce a new filled function which has different propertieswith respect to the one proposed in Lucidi and Piccialli (2002).