A problem (P): [formula omitted], where F() is the solutions set of a linear semi-infinite system, is called reducible if there exists a finite subproblem with the same value, v(P). On the other hand we say that (P) i...
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A new approach for semi-infinite programming problems is presented, which belongs to the class of successive quadratic programming (SQP) methods with trust region technique. The proposed algorithm employs the exact L ...
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A new approach for semi-infinite programming problems is presented, which belongs to the class of successive quadratic programming (SQP) methods with trust region technique. The proposed algorithm employs the exact L ∞ penalty function as a criterion function and incorporates an appropriate scheme for estimating active constraints. It is proved that the algorithm is globally convergent under some assumptions. Numerical experiments show that the algorithm is very promising in practice.
With the recent dramatic increase in available computing power, numerical optimization has become an attractive tool for the design of complex engineering systems. Yet, generalized use of numerical optimization techni...
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With the recent dramatic increase in available computing power, numerical optimization has become an attractive tool for the design of complex engineering systems. Yet, generalized use of numerical optimization techniques in design has been hindered by (i) the difficulty to translate in a faithful manner the actual design problem into any kind of rigid mathematical optimization problem, (ii) the inability of classical optimization tools to efficiently take into account the many distinctive features of optimization problems arising in a design context, and (iii) the unavailability of software tools offering to the designer a powerful as well as congenial environment supporting such capabilities. In this paper, some aspects of these questions are touched upon and avenues are suggested to address them. In particular, a recently proposed interaction driven design methodology is briefly described and numerical optimization schemes satisfying two specific requirements of many design problems are sketched. As an example, the design of a controller for a copolymerization reactor using the Maryland developed CONSOLE system is considered.
We treat semi-infinite optimization problems: Minimizep(x) subject tox ∈ ? m , anda(t,x) ≦b(t) for allt ∈T, whereT is a σ-compact topological space, andp,a,b are suitable (?∞, ∞]-valued functions on R m , respec...
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We treat semi-infinite optimization problems: Minimizep(x) subject tox ∈ ? m , anda(t,x) ≦b(t) for allt ∈T, whereT is a σ-compact topological space, andp,a,b are suitable (?∞, ∞]-valued functions on R m , respectively. Linear, convex, and quasi-convex semi-infinite programming are included in this concept. The main results of this paper are on the necessity of the compactness of the set of feasible points for (a,b), and the set of ?-optimal solutions for (p,a,b) for the (Hausdorff) upper semicontinuity of the feasible set-mapping in (a,b), and the ?-optimal solution-mapping in (p,a,b), respectively (where the parameter sets are provided with a suitable topology). Some more special results complete the paper.
For linear semi-infinite programming problems a discretization method is presented. A first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as sta...
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For linear semi-infinite programming problems a discretization method is presented. A first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems. This enables an efficient treatment of large problems with moderate storage requirements. A numerically stable Simplex-algorithm is used to solve the LP-subproblems. Numerical examples from bivariate Chebyshev approximation are presented.
We consider the problem of minimizing a nondifferentiable function that is the pointwise maximum over a compact family of continuously differentiable functions. We suppose that a certain convex approximation to the ob...
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We consider the problem of minimizing a nondifferentiable function that is the pointwise maximum over a compact family of continuously differentiable functions. We suppose that a certain convex approximation to the objective function can be evaluated. An iterative method is given which uses as successive search directions approximate solutions of semi-infinite quadratic programming problems calculated via a new generalized proximity algorithm. Inexact line searches ensure global convergence of the method to stationary points.
A globally convergent algorithm is presented for the solution of a wide class of semi-infinite programming problems. The method is based on the solution of a sequence of equality constrained quadratic programming prob...
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A globally convergent algorithm is presented for the solution of a wide class of semi-infinite programming problems. The method is based on the solution of a sequence of equality constrained quadratic programming problems, and usually has a second order convergence rate. Numerical results illustrating the method are given.
A new, simple, constraint qualification for infinite dimensional programs with linear programming type constraints is used to derive the dual program; see Theorem 3.1. Applications include a proof of the explicit solu...
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A new, simple, constraint qualification for infinite dimensional programs with linear programming type constraints is used to derive the dual program; see Theorem 3.1. Applications include a proof of the explicit solution of the best interpolation problem presented in [8].
An algorithm for the numerical solution of general systems of complex linear equations in the $l_\infty $, or Chebyshev, norm is presented. The objective is to find complex values for the unknowns so that the maximum...
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An algorithm for the numerical solution of general systems of complex linear equations in the $l_\infty $, or Chebyshev, norm is presented. The objective is to find complex values for the unknowns so that the maximum magnitude residual of the system is a minimum. The unknowns are required to satisfy certain convex constraints; in particular, bounds on the magnitudes of the unknowns are imposed. In the algorithm presented here, this problem is replaced by a linear program generated in such a way that the relative error between its solution and a solution of the original problem can be estimated. The maximum relative error can easily be made as small as desired by selecting an appropriate linear program. Order of magnitude improvements in both computation time and computer storage requirements in an implementation of the simplex algorithm to this linear program are presented. Three numerical examples are included, one of which is a complex function approximation problem.
The first part of this paper presents a new simplified version, based upon the properties of the Farkas-Minkowski systems, of the well-known relation between the Duality Theory in semi-infinite programming and a certa...
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