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L(x, g , h ) = f (x) +
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(A27)
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The dual problem associated with the primal problem in equation (A20) is then de ned as Definition A8 Dual problem: maximize g , h subject to L(x, g , h ) gm 0, m = 1, , ng + nh (A28)
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If the primal problem is convex over the search space S, then the solution to the primal problem is the vector x of the saddle point, (x , , ), of the Lagrangian in g h equation (A27), such that L(x , g , h ) L(x , , ) L(x, , ) g h g h (A29)
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The vector x that solves the primal problem, as well as the Lagrange multiplier vectors, and , can be found by solving the min-max problem, g h min max L(x, g , h )
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A Optimization Theory
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For non-convex problems, the solution of the dual problem does not coincide with the solution of the primal problem For non-convex problems, the Lagrangian is augmented by adding a penalty term, ie L (x, g , h ) = L(x, g , h ) + p(x, t) (A31)
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where > 0, L(x, g , h ) is as de ned in equation (A27), and the penalty p(x, t) is as de ned in equations (A25) and (A26) with m (t) = 1 and = 2
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A63
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Example Benchmark Problems
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A number of benchmark functions for constrained optimization are listed in this section Again, the list is not intended to be complete The objective is to provide a list of constrained problems as a starting point in evaluating algorithms for constrained optimization Constrained problem 1: Minimize the function f (x) = 100(x2 x2 )2 + (1 x1 )2 1 subject to the nonlinear constraints, x1 + x 2 0 2 2 x1 + x 2 0 with x1 [ 05, 05] and x2 10 The global optimum is x = (05, 025), with f (x ) = 025 Constrained problem 2: Minimize the function f (x) = (x1 2)2 (x2 1)2 subject to the nonlinear constraint x2 + x2 0 1 and the linear constraint x 1 + x2 2 with x = (1, 1) and f (x ) = 1 Constrained problem 3: Minimize the function
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4 13
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f (x) = 5x1 + 5x2 + 5x3 + 5x4 5
x2 j
(A34)
subject to the constraints 2x1 + 2x2 + x10 + x11 10 2x2 + 2x3 + x11 + x12 10 8x2 + x11 0 2x4 x5 + x1 0 0 2x8 x9 + x12 0 2x1 + 2x3 + x10 + x12 10 8x1 + x10 0 8x3 + x12 0 2x6 x7 + x11 0
A7 Multi-Solution Problems
with xj [0, 1] for j = 1, , 9, xj [0, 100] for j = 10, 11, 12, and x13 [0, 1] The solution is x = (1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1), with f (x ) = 15 Constrained problem 4: Maximize the function f (x) = ( nx )nx subject to the equality constraint,
nx nx
(A35)
x2 = 1 j
with xj [0, 1] The solution is x = ( 1 x , , 1 x ), with f (x ) = 1 n n Constrained problem 5: Minimize the function
105x1 75x2 35x3 25x4 15x5 10x6 05
x2 j
(A36)
subject to the constraints 6x1 + 3x2 + 3x3 + 2x4 + x5 65 10x1 + 10x3 + x6 20 0
with xj [0, 1] for j = 1, , 5, and x6 0 The best known solution is f (x) = 2130
Multi-Solution Problems
Multi-solution problems are multi-modal, containing many optima These optima may include more than one global optimum and a number of local minima, or just one global optimum together with more than one local optimum The objective of multi-solution optimization methods is to locate as many as possible of these optima A formal de nition is given in Section A71, with di erent algorithm categories listed in Section A72 Example benchmark problems are given in Section A73
A71