algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in ...
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algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.
Studies of the convergence of algorithms often revolve around the existence of a function with respect to which monotonic descent is required. In this paper, we show that under relatively lenient conditions, "sta...
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Studies of the convergence of algorithms often revolve around the existence of a function with respect to which monotonic descent is required. In this paper, we show that under relatively lenient conditions, "stage-dependent descent" (not necessarily monotonic) is sufficient to guarantee convergence. This development also provides the impetus to examine optimization algorithms. We show that one of the important avenues in the study of convergence, namely, the theory of epi-convergence imposes stronger conditions than are necessary to establish the convergence of an optimization algorithm. Working from a relaxation of epi-convergence, we introduce the notion of partial derivative-compatibility, and prove several results that permit relaxations of conditions imposed by previous approaches to algorithmic convergence. Finally, to illustrate the usefulness of the concepts, we combine stage-dependent descent with results derivable from partial derivative-compatibility to provide a basis for the convergence of a general algorithmic statement that might be used for stochastic and nondifferentiable optimization.
The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization prob...
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The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization problems (Refs. 1–3) and variational inequalities (Ref. 4). The key assumption to prove the global and strong convergence of such algorithms, as well as of most of the other algorithms proposed in the literature, is the strong monotony of the operator involved in the variational inequalities. In this paper, we consider variational inequalities defined over a product of spaces and we introduce a new property of strong nested monotony, which is weaker than the ordinary overall strong monotony generally assumed. In some sense, this new concept seems to be a minimal requirement to insure convergence of the algorithms alluded to above. A convergence theorem based on this weaker assumption is given. Application of this result to the computation of Nash equilibria can be found in another paper (Ref. 5).
In this paper, we study the convergence of proximal methods for solving pseudomonotone (in the sense of Karamardian) variational inequalities. The main result is given in the finite-dimensional case, but we show that ...
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In this paper, we study the convergence of proximal methods for solving pseudomonotone (in the sense of Karamardian) variational inequalities. The main result is given in the finite-dimensional case, but we show that we still obtain convergence in an infinite-dimensional Hilbert space under a strong pseudomonotonicity or a pseudo-Dunn assumption on the operator involved in the variational inequality problem.
In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation w...
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In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.
This paper deals with the convergence of the algorithm built on the auxiliary problem principle for solving pseudomonotone (in the sense of Karamardian) variational inequalities.
This paper deals with the convergence of the algorithm built on the auxiliary problem principle for solving pseudomonotone (in the sense of Karamardian) variational inequalities.
Each master iteration of a simplified Newton algorithm for solving a system of equations starts by computing the Jacobian matrix and then uses this matrix in the computation of p Newton steps: the first of these steps...
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Each master iteration of a simplified Newton algorithm for solving a system of equations starts by computing the Jacobian matrix and then uses this matrix in the computation of p Newton steps: the first of these steps is exact, and the other are called ''simplified''. In this paper we apply this approach to a large step path following algorithm for monotone linear complementarity problems, The resulting method generates sequences of objective values (duality gaps) that converge to zero with Q-order p + 1 in the number of master iterations, and with a complexity of O(root nL) iterations.
Conditions under which the multiplier and penalty methods converge in combination, regardless of the initial multiplier choice, are given. A duality is established between the state and costate variables. For the cont...
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Conditions under which the multiplier and penalty methods converge in combination, regardless of the initial multiplier choice, are given. A duality is established between the state and costate variables. For the control case, an interpretation in terms of the Hamilton-Jacobi partial differential equation is indicated.
Global convergence results are derived for well-known conjugate gradient methods in which the line search step is replaced by a step whose length is determined by a formula. The results include the following cases: (1...
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Global convergence results are derived for well-known conjugate gradient methods in which the line search step is replaced by a step whose length is determined by a formula. The results include the following cases: (1) The Fletcher-Reeves method, the Hestenes-Stiefel method, and the Dai-Yuan method applied to a strongly convex LC1 objective function;(2) The Polak-Ribiere method and the Conjugate Descent method applied to a general, not necessarily convex, LC1 objective function.
Three classes of optimal control problems involving second boundary value problems of parabolic type are considered. The controls are assumed to act through the forcing terms and through the initial and boundary condi...
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Three classes of optimal control problems involving second boundary value problems of parabolic type are considered. The controls are assumed to act through the forcing terms and through the initial and boundary conditions.
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