The problem of finding a parameter which satisfies a set of specifications in inequality form is sometimes referred to as the satisfycing problem. We present a family of methods for solving, in a finite number of iter...
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The problem of finding a parameter which satisfies a set of specifications in inequality form is sometimes referred to as the satisfycing problem. We present a family of methods for solving, in a finite number of iterations, satisfycing problems stated in the form of semi-infinite inequalities. These methods range from adaptive uniform discretization methods to outer approximation methods.
An algorithm for computing Kuhn-Tucker curves of one-parametric semi-infinite optimization problems p(t), tϵR is presented. Starting witii a Kuhv-Tucker point x for a fixed t an underdetermined system of nonlinear equ...
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In this paper we study the following infinite-dimensional programming problem: (P) μ?inff 0(x), subject tox∈C,f i(x)≤0,i∈I, whereI is an index set with possibly infinite cardinality andC is an infinite-dimensional...
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In this paper we study the following infinite-dimensional programming problem: (P) μ?inff 0(x), subject tox∈C,f i(x)≤0,i∈I, whereI is an index set with possibly infinite cardinality andC is an infinite-dimensional set. Zero duality gap results are presented under suitable regularity hypotheses for convex-like (nonconvex) and convex infinitely constrained program (P). Various properties of the value function of the convex-like program and its connections to the regularity hypotheses are studied. Relationships between the zero duality gap property, semicontinuity, and ε-subdifferentiability of the value function are examined. In particular, a characterization for a zero duality gap is given, using the ε-subdifferential of the value function without convexity.
The second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions. By taking account of this fact, we derive new second order necessary optimality conditions for minimiza...
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The second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions. By taking account of this fact, we derive new second order necessary optimality conditions for minimization of a sup-type function. The conditions involve an extra term besides the second derivative of the Lagrange function. Furthermore, we will comment on the relationship between the extra term and a kind of second order directional derivative of the sup-type function.
In this paper, we provide a systematic approach to the main topics in linear semi-infinite programming by means of a new methodology based on the many properties of the sub-differential mapping and the closure of a gi...
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In this paper, we provide a systematic approach to the main topics in linear semi-infinite programming by means of a new methodology based on the many properties of the sub-differential mapping and the closure of a given convex function. In particular, we deal with the duality gap problem and its relation to the discrete approximation of the semi-infinite program. Moreover, we have made precise the conditions that allow us to eliminate the duality gap by introducing a perturbation in the primal objective function. As a by-product, we supply different extensions of well-known results concerning the subdifferential mapping.
In this paper, a new method for semi-infinite programming problems with convex constraints is presented. The method generates a sequence of feasible points whose cluster points are solutions of the original problem. N...
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In this paper, a new method for semi-infinite programming problems with convex constraints is presented. The method generates a sequence of feasible points whose cluster points are solutions of the original problem. No maximization over the index set is required. Some computational results are also presented.
A computational comparison of several recent semi-infinite nonlinear programming algorithms is presented. Because of the importance of this area, there have recently been proposed a number of algorithms having a globa...
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A computational comparison of several recent semi-infinite nonlinear programming algorithms is presented. Because of the importance of this area, there have recently been proposed a number of algorithms having a global convergence property. The algorithms used in this study include a feasible direction method and some variants of successive quadratic programming methods. Robustness and relative efficiency of the algorithms are examined on the basis of the results for some standard test examples chosen from earlier literature. [ABSTRACT FROM AUTHOR]
Two extreme techniques when choosing a search direction in a linearly constrained optimization calculation are to take account of all the constraints or to use an active set method that satisfies selected constraints ...
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Two extreme techniques when choosing a search direction in a linearly constrained optimization calculation are to take account of all the constraints or to use an active set method that satisfies selected constraints as equations, the remaining constraints being ignored. We prefer an intermediate method that treats all inequality constraints with “small” residuals as inequalities with zero right hand sides and that disregards the other inequality conditions. Thus the step along the search direction is not restricted by any constraints with small residuals, which can help efficiency greatly, particularly when some constraints are nearly degenerate. We study the implementation, convergence properties and performance of an algorithm that employs this idea. The implementation considerations include the choice and automatic adjustment of the tolerance that defines the “small” residuals, the calculation of the search directions, and the updating of second derivative approximations. The main convergence theorem imposes no conditions on the constraints except for boundedness of the feasible region. The numerical results indicate that a Fortran implementation of our algorithm is much more reliable than the software that was tested by Hock and Schittkowski (1981). Therefore the algorithm seems to be very suitable for general use, and it is particularly appropriate for semi-infinite programming calculations that have many linear constraints that come from discretizations of continua.
For the problemP(λ): Maximizec T z subject toz∈Z(λ), whereZ(λ) is defined by an in general infinite set of linear inequalities, it is shown that the value-function has directional derivatives at every point \(\ba...
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For the problemP(λ): Maximizec T z subject toz∈Z(λ), whereZ(λ) is defined by an in general infinite set of linear inequalities, it is shown that the value-function has directional derivatives at every point \(\bar \lambda \)such thatP( \(\bar \lambda \) ) and its dual are both superconsistent. To compute these directional derivatives a min-max-formula, well-known in convex programming, is derived. In addition, it is shown that derivatives can be obtained more easily by a limit-process using only convergent selections of solutions ofP(λ n ), λ n → \(\bar \lambda \)and their duals.
This paper introduces a global approach to the semi-infinite programming problem that is based upon a generalisation of the ?1 exact penalty function. The advantages are that the ensuing penalty function is exact and ...
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This paper introduces a global approach to the semi-infinite programming problem that is based upon a generalisation of the ?1 exact penalty function. The advantages are that the ensuing penalty function is exact and the penalties include all violations. The merit function requires integrals for the penalties, which provides a consistent model for the algorithm. The discretization is a result of the approximate quadrature rather than an a priori aspect of the model.
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