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.
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.
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 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.
In this paper, we establish a nonlinear Lagrangian algorithm for nonlinear programming problems with inequality constraints. Under some assumptions, it is proved that the sequence of points, generated by solving an un...
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In this paper, we establish a nonlinear Lagrangian algorithm for nonlinear programming problems with inequality constraints. Under some assumptions, it is proved that the sequence of points, generated by solving an unconstrained programming, convergents locally to a Kuhn-Tucker point of the primal nonlinear programming problem.
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.
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.
In a previous paper a unified outline of some of the most successful nonlinear programming methods was presented by the author, i.e. of penalty, multiplier, sequential quadratic programming, and generalized reduced gr...
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In a previous paper a unified outline of some of the most successful nonlinear programming methods was presented by the author, i.e. of penalty, multiplier, sequential quadratic programming, and generalized reduced gradient algorithms, to illustrate their common mathematical features and to explain the different numerical performance observed in practice. By defining a general algorithmic frame for all these approaches, a global convergence result can be achieved in the sense that starting from an arbitrary initial point, a stationary solution will be approximated.
Recently developed Newton and quasi-Newton methods for nonlinear programming possess only local convergence properties. Adopting the concept of the damped Newton method in unconstrained optimization, we propose a step...
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Recently developed Newton and quasi-Newton methods for nonlinear programming possess only local convergence properties. Adopting the concept of the damped Newton method in unconstrained optimization, we propose a stepsize procedure to maintain the monotone decrease of an exact penalty function. In so doing, the convergence of the method is globalized.
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