A3 Optima Types

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Weak local minimum

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Strong local minima

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Global minimum

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Figure A1 Types of Optima for Unconstrained Problems

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Optima Types

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Solutions found by optimization algorithms are classi ed by the quality of the solution The main types of solutions are referred to as local optima or global optima For the purpose of this section, a minimization problem is assumed A global optimum (considering a minimization problem) is formally de ned as follows: Definition A1 Global minimum: The solution x F, is a global optimum of the objective function, f , if (A1) f (x ) < f (x), x F where F S The global optimum is therefore the best of a set of candidate solutions, as illustrated in Figure A1 for a minimization problem As illustrated in Figure A2, a problem may have more than one global optimum A local minimum is de ned as follows (as illustrated in Figure A1): Definition A2 Strong local minimum: The solution, x N F, is a strong N local minimum of f , if (A2) f (x ) < f (x), x N N where N F is a set of feasible points in the neighborhood of x N

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1 09 08 07 06 f(x) 05 04 03 02 01 0 0 02 04 x 06

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A Optimization Theory

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Figure A2 Problem with Multiple Global Optima, with f (x) = sin6 (5 x) Definition A3 Weak local minimum: The solution, x N F, is a weak local N minimum of f , if (A3) f (x ) f (x), x N N where N F is a set of feasible points in the neighborhood of x N

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Optimization Method Classes

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An optimization algorithm searches for an optimum solution by iteratively transforming a current candidate solution into a new, hopefully better, solution Optimization methods can be divided into two main classes, based on the type of solution that is located Local search algorithms use only local information of the search space surrounding the current solution to produce a new solution Since only local information is used, local search algorithms locate local optima (which may be a global minimum) A global search algorithm uses more information about the search space to locate a global optimum It is said that global search algorithms explore the entire search space, while local search algorithms exploit neighborhoods Optimization algorithms are further classi ed into deterministic and stochastic methods Stochastic methods use random elements to transform one candidate solution into a new solution The new point can therefore not be predicted Deterministic methods, on the other hand, do not make use of random elements Based on the problem characteristics, optimization methods are grouped in the following classes (within each of these classes further subdivision occurs based on whether local or global optima are located, and based on whether random elements are used to investigate new points in the search space):

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A5 Unconstrained Optimization unconstrained methods, used to optimize unconstrained problems;

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constrained methods, used to nd solutions in constrained search spaces; multi-objective optimization methods for problems with more than one objective to optimize; multi-solution (niching) methods with the ability to locate more than one solution; and dynamic methods with the ability to locate and track changing optima Subsequent sections discuss each of these optimization method classes

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Unconstrained Optimization

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Except for boundary constraints, unconstrained optimization problems place no restrictions on the values that can be assigned to variables of the problem The feasible space is simply the entire search space This chapter provides a formal de nition of unconstrained optimization problems in Section A51 Section A52 summarizes algorithms used in later chapters, and Section A53 gives a list of classical benchmark problems

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A51

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Problem De nition

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The general unconstrained optimization problem is de ned as Definition A4 Unconstrained optimization problem: minimize f (x), subject to x = (x1 , x2 , , xnx ) xj dom(xj )

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(A4)

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where x F = S, and dom(xj ) is the domain of variable xj For a continuous problem, the domain of each variable is R, ie xj R For an integer problem, xj Z, while dom(xi ) for a general discrete problem is a nite set of values Note that an integer problem is simply a special case of a discrete problem

A52