Progress in Global Optimization and Shape Design
In this paper, we reformulate global optimization problems in terms of boundary value problems. This allows us to introduce a new class of optimization algorithms. Indeed, many optimization methods can be seen as discretizations of initial value problems
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I3M- Universite de Montpellier II, Place Eugene Bataillon, CC051, 34095 Montpellier, France [email protected] ISTEEM - Universite de Montpellier II, Place Eugene Bataillon, 34095 Montpellier, France [email protected]
Abstract In this paper, we reformulate global optimization problems in terms of boundary value problems. This allows us to introduce a new class of optimization algorithms. Indeed, many optimization methods can be seen as discretizations of initial value problems for differential equations or systems of differential equations. We apply a particular algorithm included in the former class to the shape optimization of coastal structures.
1 Introduction Many optimization algorithms can be viewed as discrete forms of Cauchy problems for a system of ordinary differential equations in the space of control parameters [1, 2]. We will see that if one introduces an extra information on the infimum, solving global optimization problems using these algorithms is equivalent to solving Boundary Value Problems (BVP) for the same equations. A motivating idea is therefore to apply algorithms solving BVPs to perform this global optimization. In this paper we present a reformulation of global minimization problems in term of over-determined BVPs, discuss the existence of their solutions and present some algorithms solving those problems. One aim here is also to show the importance of global optimization algorithms for shape optimization. Indeed, because of excessive cost of global optimization approaches usually only local minimization algorithms are used for shape optimization of distributed systems, especially with fluids [3, 4]. Our semi-deterministic algorithm permits for global optimization of systems governed by PDEs at reasonable cost. Section 2 presents our global optimization method and mathematical background. In Section 3, the previous approach is applied to our considered optimization problem.
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D. Isebe et al.
2 Optimization Method We consider the following minimization problem: min J(x)
x∈Ωad
(1)
where J : Ωad → IR is called cost function, x is the optimization parameter and belongs to a compact admissible space Ωad ⊂ IRN , with N ∈ IN. We make the following assumptions [2]: J ∈ C 2 (Ωad , IR) and coercive. The infimum of J is denoted by Jm . 2.1 BVP Formulation of Optimization Problems Many minimization algorithms which perform the minimization of J can be seen as discretizations of continuous first or second order dynamical systems with associated initial conditions [1]. A numerical global optimization of J with one of those algorithms, called here core optimization method, is possible if the following BVP has a solution: First or second order initial value problem (2) |J(x(Z)) − Jm | < where x(Z) is the solution of the considered dynamical system found at a given finite time Z ∈ IR and is the approximation precision. In practice, when Jm is unknown, we set Jm to a lower value (for example Jm = 0 for a non-negative function J) and look for the best solution for a given comple
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