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1

Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces

Wairojjana Nopparat, Pakkaranang Nuttapol, Pholasa Nattawut

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

110--128

EN

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

2

Convergence of finite-dimensional approximations for mixed-integer optimization with differential equations

Hante Falk M., Schmidt Martin

Control and Cybernetics

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2019

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Vol. 48, No. 2

209--226

EN

We consider a direct approach to solving the mixedinteger nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer nonlinear optimization problem in finite dimensions, where the step size for discretization of the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems for gas networks.

3

On the Lipschitz continuity of the solutions of a class of elliptic free boundary problems

Lyaghfouri A.

Journal of Applied Analysis

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2008

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Vol. 14, nr 2

165-181

EN

In this article we prove local interior and boundary Lipschitz continuity of the solutions of a general class of elliptic free boundary problems in divergence form.

4

Optimal control of semilinear elliptic equation with state constraint : maximum principle for minimizing sequence, regularity, normality, sensitivity

Sumin M.

Control and Cybernetics

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2000

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Vol. 29, no 2

449-472

EN

This article deals with state constrained optimal control problem for semilinear elliptic equation in a domain Omega. The state constraint is lumped on the compactum X contained in/implied by Omega n and contains a functional parameter q in C(X ). It is shown that any minimizing approximate solution (m.a.s.) in the sense of J. Warga satisfies the pointwise maximum principle (the maximum principle for m.a.s.) if the problem is meaningful, i.e., the value of the problem is finite. It is also shown that a condition of Slater's type is sufficient for the normality in the so-called "linear-convex" problem, and the normality of the problem for some fixed value of the parameter q in C(X ) implies the Lipschitz continuity of its value function in a neighborhood of q. The paper contains illustrative examples.