Stable Barrier-Projection and Barrier-Newton Methods for Linear and Nonlinear Programming

Evtushenko, Yu. G.; Жадан, В. Г.

doi:10.1007/978-94-009-0369-2_9

Cited by 14 publications

(7 citation statements)

References 21 publications

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“…The connection Frobenius formula under the construction of generalized inverse operator matrices (also in the sense of Sherman-Morrison-Woodbury) for the operator matrices was considered in [11,12]. The Frobenius formula is widely in optimization, optimal control problems, and high-performance computing [13][14][15].…”

Section: Problem Statementmentioning

confidence: 99%

Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

2021

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The efficient construction and employment of block operators are vital for contemporary computing, playing an essential role in various applications. In this paper, we prove a generalisation of the Frobenius formula in the setting of the theory of block operators on normed spaces. A system of linear equations with the block operator acting in Banach spaces is considered. Existence theorems are proved, and asymptotic approximations of solutions in regular and irregular cases are constructed. In the latter case, the solution is constructed in the form of a Laurent series. The theoretical approach is illustrated with an example, the construction of solutions for a block equation leading to a method of solving some linear integrodifferential system.

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Section: Problem Statementmentioning

confidence: 99%

Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

2021

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“…[9]- [18] for details). Among them, Evtushenko and Zhadan [8]- [7] have studied, by using the so-called space transformation techniques, a family of numerical methods for solving optimization problems with equality and inequality constraints. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equation.…”

Section: Jin and Hongying Huangmentioning

confidence: 99%

Differential equation method based on approximate augmented Lagrangian for nonlinear programming

Jin¹,

Huang²

2020

Journal of Industrial &Amp; Management Optimization

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This paper analyzes the approximate augmented Lagrangian dynamical systems for constrained optimization. We formulate the differential systems based on first derivatives and second derivatives of the approximate augmented Lagrangian. The solution of the original optimization problems can be obtained at the equilibrium point of the differential equation systems, which lead the dynamic trajectory into the feasible region. Under suitable conditions, the asymptotic stability of the differential systems and local convergence properties of their Euler discrete schemes are analyzed, including the locally quadratic convergence rate of the discrete sequence for the second derivatives based differential system. The transient behavior of the differential equation systems is simulated and the validity of the approach is verified with numerical experiments.

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“…Evtushenko [4] studied the problem of equality constraint earlier. Yamashita [5], Evtushenko et al [6][7][8][9][10], and Pan [11] developed and improved the differential equation methods. In particular, Evtushenko and Zhadan have carried out a great deal of research on differential equation methods for nonlinear programming problems and constraint problems on general closed sets by using the stability theory of differential equations since 1973, which enriches the differential equation method of nonlinear programming problem.…”

Section: Introductionmentioning

confidence: 99%

The Second‐Order Differential Equation System with the Controlled Process for Variational Inequality with Constraints

2021

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In this paper, the variational inequality with constraints can be viewed as an optimization problem. Using Lagrange function and projection operator, the equivalent operator equations for the variational inequality with constraints under the certain conditions are obtained. Then, the second-order differential equation system with the controlled process is established for solving the variational inequality with constraints. We prove that any accumulation point of the trajectory of the second-order differential equation system is a solution to the variational inequality with constraints. In the end, one example with three kinds of different cases by using this differential equation system is solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the controlled process for solving the variational inequality with constraints.

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Stable Barrier-Projection and Barrier-Newton Methods for Linear and Nonlinear Programming

Cited by 14 publications

References 21 publications

Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

Differential equation method based on approximate augmented Lagrangian for nonlinear programming

The Second‐Order Differential Equation System with the Controlled Process for Variational Inequality with Constraints

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