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The convex, non-uniform Pareto front in Figure A6(b) for MOP f(x) = f1 (x) = f2 (x) = where g(x) = 1 + 9 nx 1 (f1 (x), f2 (x)) x1 g(x)(1 f1 (x)/g(x))
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A Optimization Theory
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(a) Convex, Uniform Front for Equation (A49)
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(b) Convex, Non-uniform Front for Equation (A50)
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(e) Discrete Convex Front for Equation (A53)
Figure A6 Example Pareto-Optimal Fronts
A9 Dynamic Optimization Problems
The concave Pareto front in Figure A6(c) for the MOP of Figure A6(b), but with (A51) f2 (x) = g(x)(1 (f1 (x)/g(x))2 ) The partially concave and partially convex Pareto front in Figure A6(d) for the MOP of Figure A6(b), but with f2 (x) = g(x)(1
f1 (x)/g(x) (f1 (x)/g(x))4)
(A52)
The discrete, convex Pareto front in Figure A6(e) for the MOP of Figure A6(b), but with f2 (x) = g(x)(1 f1 (x)/g(x) (f1 (x)/g(x)) sin(10 f1 (x))) (A53)
The objective when solving a MOP is to approximate the true Pareto-optimal front, and then to select the solution that represents the best trade-o (for problems which, in the end, require only one solution) To nd the exact true Pareto-optimal front (ie to nd all the Pareto-optimal solutions in F) is usually computationally prohibitive The task is therefore reduced to nding an approximation to the true Pareto front such that the distance to the Pareto front is minimized, the set of non-dominated solutions, ie the Pareto-optimal set, is as diverse as possible, and already found non-dominated solutions are maintained The task of nding an approximation to the true Pareto front is therefore in itself a MOP, where the rst objective ensures an accurate approximation, and the second objective ensures that the entire Pareto front is covered
Dynamic Optimization Problems
Dynamic optimization problems have objective functions that change over time Such changes in objective function cause changes in the position of optima, and the characteristics of the search space Existing optima may disappear while new optima may appear This chapter provides a formal de nition of a dynamic problem in Section A91, and lists di erent types of dynamic problems in Section A92 Example benchmark problems are given in Section A93
A91
De nition
A dynamic optimization problem is formally de ned as
576 Definition A16 Dynamic optimization problem: minimize f (x, subject to (t)),
A Optimization Theory
x = (x1 , , xnx ), (t) = ( 1 (t), , gm (x) 0, m = 1, , ng hm (x) = 0, m = ng + 1, , ng + nh xj dom(xj )
(A54)
where (t) is a vector of time-dependent objective function control parameters The objective is to find x (t) = min f (x, (t)) (A55)
where x (t) is the optimum found at time step t The goal of an optimization algorithm for dynamic environments is then to locate an optimum and to track its trajectory as closely as possible
A92
Dynamic Environment Types
In order to track the optimum over time, the optimization algorithm needs to detect and track changes The environment may change on any timescale, referred to as temporal severity Changes can be continuously spread over time, at irregular time intervals or periodically Due to these changes, the position of an optimum may change by any amount, referred to as spatial severity Eberhart et al de nes three types of dynamic environments [228, 385]: Type I environments, where the location of the optimum in problem space is subject to change The change in the optimum, x (t) is quanti ed by the severity parameter, , which measures the jump in location of the optimum Type II environments, where the location of the optimum remains the same, but the value, f (x (t)), of the optimum changes Type III environments, where both the location of the optimum and its value changes The changes in the environment, as caused by the control parameters, can be in one or more of the dimensions of the problem If the change is in all the dimensions, then for type I environments, the change in optimum is quanti ed by I, where I is the unit vector Examples of these types of dynamic environments are illustrated in Figures A7 and A8 Figure A7 illustrates the dynamic function,
f (x,
(t)) =