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A new simultaneous subgradient projection algorithm for solving a multiple-sets split feasibility problem

Yazheng Dang, Yan Gao (2014)

Applications of Mathematics

In this paper, we present a simultaneous subgradient algorithm for solving the multiple-sets split feasibility problem. The algorithm employs two extrapolated factors in each iteration, which not only improves feasibility by eliminating the need to compute the Lipschitz constant, but also enhances flexibility due to applying variable step size. The convergence of the algorithm is proved under suitable conditions. Numerical results illustrate that the new algorithm has better convergence than the...

A nonmonotone line search for the LBFGS method in parabolic optimal control problems

Omid Solaymani Fard, Farhad Sarani, Akbar Hashemi Borzabadi, Hadi Nosratipour (2019)

Kybernetika

In this paper a nonmonotone limited memory BFGS (NLBFGS) method is applied for approximately solving optimal control problems (OCPs) governed by one-dimensional parabolic partial differential equations. A discretized optimal control problem is obtained by using piecewise linear finite element and well-known backward Euler methods. Afterwards, regarding the implicit function theorem, the optimal control problem is transformed into an unconstrained nonlinear optimization problem (UNOP). Finally the...

A nonsmooth version of the univariate optimization algorithm for locating the nearest extremum (locating extremum in nonsmooth univariate optimization)

Marek Smietanski (2008)

Open Mathematics

An algorithm for univariate optimization using a linear lower bounding function is extended to a nonsmooth case by using the generalized gradient instead of the derivative. A convergence theorem is proved under the condition of semismoothness. This approach gives a globally superlinear convergence of algorithm, which is a generalized Newton-type method.

A note on a class of equilibrium problems with equilibrium constraints

Jiří V. Outrata (2004)

Kybernetika

The paper concerns a two-level hierarchical game, where the players on each level behave noncooperatively. In this way one can model eg an oligopolistic market with several large and several small firms. We derive two types of necessary conditions for a solution of this game and discuss briefly the possibilities of its computation.

A note on direct methods for approximations of sparse Hessian matrices

Miroslav Tůma (1988)

Aplikace matematiky

Necessity of computing large sparse Hessian matrices gave birth to many methods for their effective approximation by differences of gradients. We adopt the so-called direct methods for this problem that we faced when developing programs for nonlinear optimization. A new approach used in the frame of symmetric sequential coloring is described. Numerical results illustrate the differences between this method and the popular Powell-Toint method.

A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints

René Henrion, Jiří Outrata, Thomas Surowiec (2010)

Kybernetika

In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity...

A primal-dual integral method in global optimization

Jens Hichert, Armin Hoffmann, Huan Xoang Phú, Rüdiger Reinhardt (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Using the Fenchel conjugate F c of Phú’s Volume function F of a given essentially bounded measurable function f defined on the bounded box D ⊂ Rⁿ, the integral method of Chew and Zheng for global optimization is modified to a superlinearly convergent method with respect to the level sequence. Numerical results are given for low dimensional functions with a strict global essential supremum.

A second order η -approximation method for constrained optimization problems involving second order invex functions

Tadeusz Antczak (2009)

Applications of Mathematics

A new approach for obtaining the second order sufficient conditions for nonlinear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order η -approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order η -approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear...

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