A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing ...
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A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing sequence for this problem are formally constructed from elements of minimizing sequences for its augmented Lagrangian with values of dual variables chosen by applying the Tikhonov stabilization method in the course of solving the corresponding modified dual problem. A sequential Kuhn-Tucker theorem in nondifferential form is proved in terms of minimizing sequences and augmented Lagrangians. The theorem is stable with respect to errors in the initial data and provides a necessary and sufficient condition on the elements of a minimizing sequence. It is shown that the structure of the augmented Lagrangian is a direct consequence of the generalized differentiability properties of the value function in the problem. The proof is based on a "nonlinear" version of the dual regularization method, which is substantiated in this paper. An example is given illustrating that the formal construction of a minimizing sequence is unstable without regularizing the solution of the modified dual problem.
In this paper, we propose a class of penalty methods with stochastic approximation for solving stochastic nonlinear programming problems. We assume that only noisy gradients or function values of the objective functio...
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In this paper, we propose a class of penalty methods with stochastic approximation for solving stochastic nonlinear programming problems. We assume that only noisy gradients or function values of the objective function are available via calls to a stochastic first-order or zeroth-order oracle. In each iteration of the proposed methods, we minimize an exact penalty function which is nonsmooth and nonconvex with only stochastic first-order or zeroth-order information available. Stochastic approximation algorithms are presented for solving this particular subproblem. The worst-case complexity of calls to the stochastic first-order (or zeroth-order) oracle for the proposed penalty methods for obtaining an epsilon-stochastic critical point is analyzed.
In this article, we consider a lower order penalty function and its epsilon-smoothing for an inequality constrained nonlinear programming problem. It is shown that any strict local minimum satisfying the second-order ...
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In this article, we consider a lower order penalty function and its epsilon-smoothing for an inequality constrained nonlinear programming problem. It is shown that any strict local minimum satisfying the second-order sufficiency condition for the original problem is a strict local minimum of the lower order penalty function with any positive penalty parameter. By using an epsilon-smoothing approximation to the lower order penalty function, we get a modified smooth global exact penalty function under mild assumptions.
In this paper a new continuously differentiable exact penalty function is introduced for the solution of nonlinear programming problems with compact feasible set. A distinguishing feature of the penalty function is th...
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In this paper a new continuously differentiable exact penalty function is introduced for the solution of nonlinear programming problems with compact feasible set. A distinguishing feature of the penalty function is that it is defined on a suitable bounded open set containing the feasible region and that it goes to infinity on the boundary of this set. This allows the construction of an implementable unconstrained minimization algorithm, whose global convergence towards Kuhn-Tucker points of the constrained problem can be established.
In this paper,an optimality condition for nonlinear programming problems with box constraints is given by using linear transformation and Lagrange interpolating *** on this condition,two new local optimization method...
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In this paper,an optimality condition for nonlinear programming problems with box constraints is given by using linear transformation and Lagrange interpolating *** on this condition,two new local optimization methods are *** solution points obtained by the new local optimization methods can improve the Karush–Kuhn–Tucker(KKT)points in *** global optimization methods then are proposed by combining the two new local optimization methods with a filled function *** numerical examples are reported to show the effectiveness of the proposed methods.
Optimal repair-replacement problem is an important aspect of economic decision making at the firm and aggregate levels. In this paper, we extend the continuous time optimal replacement model in the firm under technolo...
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Optimal repair-replacement problem is an important aspect of economic decision making at the firm and aggregate levels. In this paper, we extend the continuous time optimal replacement model in the firm under technological progress by considering the possibility of repairing/replacing the machines during their lifetime period. In our model, two possible decisions can be recognized by the managers in which the machines are repaired under the efficiency condition or replaced under the availability of technological progress in the firm. As a special case, we restrict the model to the more real case in which all the growth, purchase price and repair cost functions are assumed to be in the exponential form. The solvability of the model in this case is also discussed. (C) 2013 Elsevier B.V. All rights reserved.
Optimality conditions are derived for a nonlinear program in which a support function appears in the objectives as well as in each constraint function. Wolfe and Mond-Weir type duals to this program are presented and ...
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Optimality conditions are derived for a nonlinear program in which a support function appears in the objectives as well as in each constraint function. Wolfe and Mond-Weir type duals to this program are presented and various duality results are established under suitable convexity and generalized convexity assumptions. Special cases that often occur in the literature are those in which a support function is the square root of a positive semi-definite quadratic form or an Lp norm. It is pointed out that these special cases can easily be generated from our results.
Solving ventilation networks of natural air splitting is a classical problem in mine ventilation. A common approach to this problem is to formulate it based on Kirchhoff's voltage and current laws and obtain the s...
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Solving ventilation networks of natural air splitting is a classical problem in mine ventilation. A common approach to this problem is to formulate it based on Kirchhoff's voltage and current laws and obtain the solution by an iterative technique known as the Hardy Cross method. In this paper, it is shown that the problem can be formulated and analysed as an unconstrained optimization (minimization) problem. The computational experience with the method of conjugate gradients is also discussed.
A unified theory of duality in linear and nonlinear programming can be easily derived from the perspective of implied constraints, that is, constraints implied by the given constraints of the problem. The key idea is ...
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A unified theory of duality in linear and nonlinear programming can be easily derived from the perspective of implied constraints, that is, constraints implied by the given constraints of the problem. The key idea is to construct special types of implied constraints which will bound the objective function. This simple idea motivates a complete, concise and clear development of a unified theory of duality which seeks the tightest bound on the primal objective function. The derivation is intuitive and geometric and leads to a more general formulation of duality in nonlinear programming than the formulations known so far. Duality theorems are shown to hold for this formulation in general. To solve the dual, the set of implied constraints on the primal objective function needs to be generated. A duality gap exists if this set is only partially generated in a dual formulation. This is implicity the case in formulations such as the Lagrangian and surrogate duality. In linear programming, this set can be generated linearly resulting in the constraints of the dual problem. This formulation is also equivalent to a special conjugate function. The conjugate function approach to duality is formulated more for the perturbations in the parameters of the problem and has not developed this fact.
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