Multi-Objective Optimization in Visual Studio .NET

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Multi-Objective Optimization
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Many real-world problems require the simultaneous optimization of a number of objective functions Some of these objectives may be in con ict with one another For example, consider nding optimal routes in data communications networks, where the objectives may include to minimize routing cost, to minimize route length, to minimize congestion, and to maximize utilization of physical infrastructure There is an important trade-o between the last two objectives: minimization of congestion is
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A Optimization Theory
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achieved by reducing the utilization of links A reduction in utilization, on the other hand, means that infrastructure, for which high installation and maintenance costs are incurred, is under-utilized This chapter provides a theoretical overview of multi-objective optimization (MOO), focusing on de nitions that are needed in later chapters The objective of this chapter is by no means to give a complete treatment of MOO The reader can nd more in-depth treatments in [150, 195] Section A81 de nes the multi-objective problem (MOP), and discusses the meaning of an optimum in terms of MOO Section A82 summarizes weight aggregation approaches to solve MOPs Section A83 provides de nitions of Pareto-optimality and dominance, and a lists a few example problems
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A81
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Multi-objective Problem
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Let S Rnx denote the nx -dimensional search space, and F S the feasible space With no constraints, the feasible space is the same as the search space Let x = (x1 , x2 , , xnx ) S, referred to as a decision vector A single objective function, fk (x), is de ned as fk : Rnx R Let f(x) = (f1 (x), f2 (x), , fnk (x)) O Rnk be an objective vector containing nk objective function evaluations; O is referred to as the objective space The search space, S is also referred to as the decision space Using the notation above, the multi-objective optimization problem is de ned as: Definition A10 Multi-objective problem: minimize f(x) subject to gm (x) 0, m = 1, , ng hm (x) = 0, m = ng + 1, , ng + nh x [xmin , xmax ]nx
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(A43)
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In equation (A43), gm and hm are respectively the inequality and equality constraints, while x [xmin , xmax ] represents the boundary constraints Solutions, x , to the MOP are in the feasible space, ie all x F The meaning of an optimum has to be rede ned for MOO In terms of uni-objective optimization (UOO) where only one objective is optimized, a local optimum and global optimum is as de ned in Section A3 In terms of MOO, the de nition of optimality is not that simple The main problem is the presence of con icting objectives, where improvement in one objective may cause a deterioration in another objective For example, maximization of the structural stability of a mechanical structure may cause an increase in costs, working against the additional objective to minimize costs Tradeo s exist between such con icting objectives, and the task is to nd solutions that balance these trade-o s Such a balance is achieved when a solution cannot improve any objective without degrading one or more of the other objectives These solutions are referred to as non-dominated solutions, of which many may exist
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A8 Multi-Objective Optimization
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The objective when solving a MOP is therefore to produce a set of good compromises, instead of a single solution This set of solutions is referred to as the non-dominated set, or the Pareto-optimal set The corresponding objective vectors in objective space are referred to as the Pareto front The concepts of dominance and Pareto-optimality are de ned in the next section
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A82
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Weighted Aggregation Methods
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One of the simplest approaches to deal with MOPs, is to de ne an aggregate objective function as a weighted sum of the objectives Uni-objective optimization algorithms can then be applied, without any changes to the algorithm, to nd optimum solutions For aggregation methods, the MOP is rede ned as
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k minimize k=1 k fk (x) subject to gm (x) 0, m = 1, , ng hm (x) = 0, m = ng + 1, , ng + nh x [xmin , xmax ]nx k 0, k = 1, , nk
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(A44)
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