R. Reemtsen, "Discretization methods for the solution of semiinfinite programming problems," Journal of Optimization Theory and Applications, vol. 71, no. 1, pp. 85--103, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... conclusion of Theorem 3 should be modified as follows: For a su#cient large iteration number i 0 it holds M i,s T (u) 8i i 0 , 1 s s(i) The deleting procedure, defined by (14) 15) is compatible with grid selection strategies, used in discretization methods for SIP (see [3] [9] and references therein) Namely, these strategies can be applied in order to clean the grid M i,s in the same manner as they are usually used for cleaning a given finite grid. 24 A. Kaplan and R. Tichatschke 4 Rate of convergence In order to prove linear convergence of the sequence # i,s ....

R. Reemtsen. Discretization methods for the solution of semi-infinite programming problems. J. Opt. Theory Appl., 71:85--103, 1991.

.... problems apprx.P[Y n ] n= 1,2, N; M: the number of elements in apprx.P[Y N ] C: the number of elements in Y n ; v # : v # : max y#Y # #(x N ,y) F(y) where Y # is an equispace grid, Y # # Y 0 ; v: v : max y#YN #(x N ,y) F(y) We consider the following examples (see [11, 20, 21]) Example 8.1. Data: F (y 1 ,y 2 ) log(y 1 y 2 )siny 1 , Y 0 = 0,1] 1, 2.5] z i ( asin(8.1) Parameters: # =0.1, #=0.01, # =0.3,#=0.7, # n= max(# 0 (1.2) n , 10 3 ) n =1,2, # n =# 0 (1.2) n , n =1,2, # 0 =2. Results: See Tables 8.1 and 8.2 ( Y # = 10000) ....

....on the simplest test problem from Example 8.1. Certainly, advantages of SMETH.ACTIV.apprx over the nonactivated SMETH.ACTIV.apprx are already clear after the considerations of section 4. Regarding methods based on active search procedures we consider results of numerical experiments reported in [21]. Recall that Reemtsen s method involves a refined active search of relevant parameters and at the nth iteration to form a simpler linear problem apprx.P[Y n 1 ] it solves the following discrete inner maximization problem: #(x n ,y) F(y) # max y# Yn , where Y n is an equispace grid, Y ....

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R. REEMTSEN, Discretization methods for the solution of semi-infinite programming problems, J. Optim. Theory Appl., 71 (1991), pp. 85--103.

....the problem (SI) in addition to some algorithms and applications, see [9] as well as the other papers in the present volume. Many globally convergent algorithms designed to solve (SI) 2 Chapter 1 rely on approximating Phi(x) by using progressively finer discretizations of [0; 1] see, e.g. [5, 7, 8, 16, 18, 19, 20, 23]) Specifically, such algorithms generate a sequence of problems of the form minimize f(x) subject to OE(x; 0; 8 2 Xi; DSI) where Xi ae [0; 1] is a (presumably large) finite set. For example, given q 2 IN, one could use the uniform discretization Xi Delta = ae 0; 1 q ; q ....

R. Reemtsen. Discretization methods for the solution of semi-infinite programming problems. J. Optim. Theory Appl., 71:85--103, 1991.

....from the fact that the discretized problem has less constraints than the SIP problem [17, p. 15] By the duality result in [20, Theorem 6.11] and continuity of the trigonometric functions in (3.3) E( a) V(PM ) can be made arbitrarily close to V(PSIP ) by choosing M large enough [17, p. 113] [35]. Of particular interest is the performance of the solution a relative to the optimal solution V(PSIP ) For Algorithm I, Theorem 6.1 below provides an upper bound on the error due to discretization. This bound tends to zero as (N=M) 2 . For M = 20N the error is less than 1.3 . The numerical ....

....coding gain which provide a measure of confidence in the result of the algorithm. If the bounds are not close enough, the discretization should be made finer. This observation points out to the use of more sophisticated discretization techniques. There is ample literature on this subject, see [35] for a discussion of adaptive discretization techniques. When M=N is very large, the simplex algorithm experiences near degeneracy. Indeed, clustered zeroes give rise to nearly dependent linear constraints [17, pp. 136 137] 35] For Algorithm I, clustered zeroes have the additional consequence ....

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R. Reemtsen, "Discretization Methods for the Solution of Semi--Infinite Programming Problems," J. Opt. Theory and Applic., Vol. 71, No. 1, Oct. 1991.

....1] i.e. substituting for (SI) the problems (DSI) minimize f(x) s.t. OE(x; 0 8 2 Omega with, for instance, Omega = f0; 1 q ; 2 q ; Delta Delta Delta ; q Gamma 1) q ; 1g; where q, a positive integer, is progressively increased (see, e.g. 10] 13] 16] 27] 31] 32] 34] [38]) The overall performance of these algorithms depends heavily on the performance at each discretization level, especially when q becomes large. Problem (DSI) involves finitely many smooth constraints and thus in principle can be solved by classical constrained optimization techniques. Yet ....

R. REEMTSEN, Discretization methods for the solution of semi-infinite programming problems, J. Optim. Theory Appl., 71 (1991), pp. 85--103.

....from the fact that the discretized problem has less constraints than the SIP problem [17, p. 15] By the duality result in [20, Theorem 6.11] and continuity of the trigonometric functions in (3.3) E( a) V(PM ) can be made arbitrarily close to V(PSIP ) by choosing M large enough [17, p. 113] [34]. Of particular interest is the performance of the solution a relative to the optimal solution V(PSIP ) For Algorithm I, Theorem 6.1 below provides an upper bound on the error due to discretization. This bound tends to zero as (N=M) 2 . For M = 20N the error is less than 1.3 . The numerical ....

....coding gain which provide a measure of confidence in the result of the algorithm. If the bounds are not close enough, the discretization should be made finer. This observation points out to the use of more sophisticated discretization techniques. There is ample literature on this subject, see [34] for a discussion of adaptive discretization techniques. When M=N is very large, the simplex algorithm experiences near degeneracy. Indeed, clustered zeroes give rise to nearly dependent linear constraints [17, pp. 136 137] 34] For Algorithm I, clustered zeroes have the additional consequence ....

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R. Reemtsen, "Discretization Methods for the Solution of Semi--Infinite Programming Problems, " J. Opt. Theory and Applic., Vol. 71, No. 1, Oct. 1991.

....this solution. An extension of the above convergence result guarantees convergence of solutions x i of problems P [D i ] i 2 N 0 ; where D 0 : Y 0 , the set D i 1 satisfies D i [ Phi y i Psi D i 1 Y i 1 ; and y i is a point with g(x i ; y i ) max y2Y i 1 g(x i ; y) [40], 42] 43] A variant of this statement concerning also nonlinear problems is derived in [30, p.464] where x i only needs to be a certain 2 (approximate) stationary point of P [D i ] Rules which allow to drop some of the constraints in P [D i ] were given in [8] 26] But, the size of jD i ....

....) is bounded for some x F 2 F (Y ) then each problem P [D k ] has a global solution. It can be shown that this algorithm stops after finitely many iterations with a global solution x k 2 F (Y i ) of P [Y i ] provided that (D k 1 ) D k ) is true only for finitely many k 2 N in succession ([40]) The latter is guaranteed, for example, if the solution of problem P [D k ] is unique ( 40] which is almost always the case on a computer. In practice, such solution should be computed via solution of the dual problem for P [D k ] since, in that way, P [D k 1 ] can be solved very efficiently ....

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Reemtsen, R.: `Discretization methods for the solution of semi-infinite programming problems', J. Optim. Theory Appl. 71 (1991), 85--103.

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R. Reemtsen, "Discretization methods for the solution of semiinfinite programming problems," Journal of Optimization Theory and Applications, vol. 71, no. 1, pp. 85--103, 1991.

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