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Partitioning procedures for solving mixed-variables programming problems

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Numerische Mathematik 1962年第1期4卷 238-252页

作者： Benders, J.F. Koninklijke/Shell-Laboratorium Amsterdam Badhuisweg 3 Amsterdam-Noord Netherlands

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A quadratically-convergent algorithm for general nonlinear programming problems

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Mathematical programming 1972年第1期3卷 145-156页

作者： Robinson, Stephen M. University of Wisconsin Madison Wisconsin United States

This paper presents an algorithm for solving nonlinearly constrained nonlinear programming problems. The algorithm reduces the original problem to a sequence of linearly-constrained minimization problems, for which efficient algorithms are available. A convergence theorem is given which states that if the process is started sufficiently close to a strict second-order Kuhn-Tucker point, then the sequence produced by the algorithm exists and converges R-quadratically to that point. © 1972 The Mathematical programming Society.

关键词： Mathematical Method programming problem Minimization problem Original problem Nonlinear programming

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A method for sequential absolute minimization in mathematical programming problems

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Cybernetics 1977年第2期13卷 143-151页

作者： Ivanov, V.V. Lyudvichenko, V.A.

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Penalty function versus non-penalty function methods for constrained nonlinear programming problems

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Mathematical programming 1971年第1期1卷 217-238页

作者： McCormick, Garth P. Research Analysis Corporation McLean Virginia United States

The relative merits of using sequential unconstrained methods for solving: minimize f(x) subject to gi(x) ≥ 0, i = 1, ⋯, m, hj(x) = 0, j = 1, ⋯, p versus methods which handle the constraints directly are explored. No... 详细信息

The relative merits of using sequential unconstrained methods for solving: minimize f(x) subject to g_i(x) ≥ 0, i = 1, ⋯, m, h_j(x) = 0, j = 1, ⋯, p versus methods which handle the constraints directly are explored. Nonlinearly constrained problems are emphasized. Both classes of methods are analyzed as to parameter selection requirements, convergence to first and second-order Kuhn-Tucker Points, rate of convergence, matrix conditioning problems and computations required. © 1971 North-Holland Publishing Company.

关键词： Mathematical Method programming problem Function Method Nonlinear programming Penalty Function

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Constant discrete programming problems on substitution sets

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Cybernetics 1976年第5期12卷 786-793页

作者： Gabovich, E.Ya.

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A method of approximate solution of discrete programming problems

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Cybernetics 1978年第3期14卷 395-401页

作者： Bykov, V.N. Sergienko, I.V.

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A dual approach to solving nonlinear programming problems by unconstrained optimization

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Mathematical programming 1973年第1期5卷 354-373页

作者： Rockafellar, R.Tyrrell University of Washington Seattle Washington United States

Several recent algorithms for solving nonlinear programming problems with equality constraints have made use of an augmented "penalty" Lagrangian function, where terms involving squares of the constraint functions are added to the ordinary Lagrangian. In this paper, the corresponding penalty Lagrangian for problems with inequality constraints is described, and its relationship with the theory of duality is examined. In the convex case, the modified dual problem consists of maximizing a differentiable concave function (indirectly defined) subject to no constraints at all. It is shown that any maximizing sequence for the dual can be made to yield, in a general way, an asymptotically minimizing sequence for the primal which typically converges at least as rapidly. © 1973 The Mathematical programming Society.

关键词： Mathematical Method programming problem Equality Constraint Nonlinear programming Dual problem

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A language for nonlinear programming problems

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Mathematical programming 1972年第1期2卷 176-206页

作者： Pugh, Robert E. Research Analysis Corporation McLean United States

An algebraic-like language for nonlinear programming problems is described. The rationale for the computation of the function values, gradients, and sēcond partial derivatives of the functions from their algebraic representation is developed. Each function is translated into an explicit "factorable" form or hierarchical representation which is used interpretively to compute the function value, gradient, and second partials of the function at each point for which such values are required. Computational efficiency is achieved by computing the matrix of second partials as the sum of a set of vector outer products, the vectors having resulted from the gradient computation, plus a diagonal matrix. An experimental computer program which implements the language and ties it to SUMT is described. In the experience with this program the computer times required have ranged from 4 to 30 times those times required by computer solutions to the same problems by using analyst-prepared programs to compute the function values, gradients, and second partial derivatives. A program based on a compiler approach to implementing the language, rather than the interpretative approach of the experimental program, will probably result in computer times between one and two times those required by using analyst-prepared programs. © 1972 North-Holland Publishing Company.

关键词： Computer Time Computer Program Partial Derivative Mathematical Method programming problem

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The method of differential descent for the solution of convex programming problems, using the principle of functional control

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Cybernetics 1976年第1期12卷 101-105页

作者： Boichuk, L.M.

The algorithm of differential descent obtained here, defined by expressions (3), (5), (9a), (10a), (12), allows a stable motion of the image point of the system to be realized from arbitrary initial, position in the feasible region. This motion is controlled by the form of the prescribed, in particular, monotonic, process of treating the errors of the completed constraints in the form of equalities (4). Within the feasible region the minimized function (1) varies to the value corresponding to the point of conditional extremum. This motion is controlled by the form of the process that, according to condition (13), is also monotonic. By virtue of the stability of these processes exact incidence on the feasible region is attainable, and within it, motion to the extremum point. The duration of these processes is defined by the coefficients of the required differential equations (6) and the coefficient γ₀, entering into expression (13) for the rate of change of the optimized function in the feasible region, where the coefficients can be adjusted independently. A program has been written for the present algorithm in ALGOL 60 [20]. This program has been certified on the solution of several nonlinear programming problems. © 1977 Plenum Publishing Corporation.

关键词： Differential Equation Operating System Artificial Intelligence System Theory programming problem

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Cut-off method with space extension in convex programming problems

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Cybernetics 1977年第1期13卷 94-96页

作者： Shor, N.Z.

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