multiobjMATLAB - 3 Standard Algorithms Multiobjective...

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3 Standard Algorithms Multiobjective Optimization In this section. .. “Introduction” on page 3-42 “Weighted Sum Method” on page 3-44 “Epsilon-Constraint Method” on page 3-46 “Goal Attainment Method” on page 3-48 “Algorithm Improvements for the Goal Attainment Method” on page 3-49 Introduction The rigidity of the mathematical problem posed by the general optimization formulation given in GP (Equation 3-1) is often remote from that of a practical design problem. Rarely does a single objective with several hard constraints adequately represent the problem being faced. More often there is a vector of objectives that must be traded off in some way .Therelativeimportanceoftheseobjectives is not generally known until the system’s best capabilities are determined and tradeoffs between the objectives fully understood. As the number of objectives increases, tradeoffs are likely to become complex and less easily quantified. There is much reliance on the intuition of the designer and his or her ability to express preferences throughout the optimization cycle. Thus, requirements for a multiobjective design strategy are to enable a natural problem formulation to beexpressed ,yettobeabletosolvetheproblemandenterpreferencesintoa numerically tractable and realistic design problem. Multiobjective optimization is concerned with the minimization of a vector of objectives F ( x
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Multiobjective Optimization (3-44) Note that, because F ( x ) is a vector, if any of the components of F ( x )are competing ,thereisnouniquesolutiontothisproblem .Instead ,theconcept of noninferiority [41] (also called Pareto optimality [4] and [6]) must be used to characterize the objectives. A noninferior solution is one in which an improvement in one objective requires a degradation of another. To define this concept more precisely, consider a feasible region, ,intheparameter space x is an element of the n-dimensional real numbers that satisfies all the constraints, i.e., (3-45) subject to This allows definition of the corresponding feasible region for the objective function space . (3-46) The performance vector, F ( x ), maps parameter space into objective function space, as represented in two dimensions in the figure below.
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3 Standard Algorithms Definition: point is a noninferior solution if for some neighborhood of there does not exist a such that and (3-47) In the two-dimensional representation of the figure below, the set of noninferior solutions lies on the curve between C and D .Po ints A and B represent specific noninferior points.
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multiobjMATLAB - 3 Standard Algorithms Multiobjective...

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