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  • nonlinear programming  (101)
  • maximum principle  (60)
  • linear programming  (45)
  • 1
    ISSN: 1573-2878
    Keywords: Convex programming ; semidefinite programming ; linear matrix inequalities ; linear programming ; constraint-aggregation method ; minimum-penalty path ; exterior path-following methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable but convex. It covers several standard problems, such as linear and quadratic programming, and has many applications in engineering. In this paper, we introduce the notion of minimal-penalty path, which is defined as the collection of minimizers for a family of convex optimization problems, and propose two methods for solving the problem by following the minimal-penalty path from the exterior of the feasible set. Our first method, which is also a constraint-aggregation method, achieves the solution by solving a sequence of linear programs, but exhibits a zigzagging behavior around the minimal-penalty path. Our second method eliminates the above drawback by following efficiently the minimum-penalty path through the centering and ascending steps. The global convergence of the methods is proved and their performance is illustrated by means of an example.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1573-2878
    Keywords: Stochastic control ; linear programming ; numerical comparisons ; numerical verification ; moments ; bounded follower
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We provide two approaches to the numerical analysis of stochastic control problems. The analyses rely on linear programming formulations of the control problem and allow numerical comparison between controls and numerical verification of optimality. The formulations characterize the processes through the moments of the induced occupation measures. We deal directly with the processes rather than with some approximation to the processes. Excellent software is readily available, since the computations involve finite-dimensional linear programs.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1573-2878
    Keywords: Multicriteria optimization ; vector-valued optimization ; maximum principle ; necessary conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control problem with vector-valued cost is considered. The satisfaction of necessary conditions for this problem is related to the satisfaction of such conditions for the problems with individual (component) scalar-valued costs.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1573-2878
    Keywords: Newton-Raphson method ; quasilinearization method ; mathematical programming ; nonlinear programming ; quadratically convergent algorithms.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The usual approach to Newton's method for mathematical programming problems with equality constraints leads to the solution of linear systems ofn +m equations inn +m unknowns, wheren is the dimension of the space andm is the number of constraints. Moreover, these linear systems are never positive definite. It is our feeling that this approach is somewhat artificial, since in the unconstrained case the linear systems are very often positive definite. With this in mind, we present an alternate Newton-like approach for the constrained problem in which all the linear systems are of order less than or equal ton. Furthermore, when the Hessian of the Lagrangian at the solution is positive definite (a situation frequently occurring), all our systems will be positive definite. Hence, in all cases, our Newton-like method offers greater numerical stability. We demonstrate that the convergence properties of this Newton-like method are superior to those of the standard approach to Newton's method. The operation count for the new method using Gaussian elimination is of the same order as the operation count for the standard method. However, if the Hessian of the Lagrangian at the solution is positive definite and we use Cholesky decomposition, then the order of the operation count for the new method is half that for the standard approach to Newton's method. This theory is generalized to problems with both equality and inequality constraints.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; penalty-function methods ; convergence rate ; method of centers
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Convergence of a method of centers algorithm for solving nonlinear programming problems is considered. The algorithm is defined so that the subproblems that must be solved during its execution may be solved by finite-step procedures. Conditions are given under which the algorithm generates sequences of feasible points and constraint multiplier vectors that have accumulation points satisfying the Fritz John or the Kuhn-Tucker optimality conditions. Under stronger assumptions, linear convergence rates are established for the sequences of objective function, constraint function, feasible point, and multiplier values.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Controllability ; maximum principle ; nonlinear systems ; control theory ; bounded-state problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A controllability minimum principle and two associated transversality conditions are presented, dealing with the controllability of nonlinear systems. The theorems represent necessary conditions for a control function to generate a system path which lies in the boundary of the set of points that are controllable to a target. The theorems presented here are controllability counterparts to Pontryagin's maximum principle, and undoubtedly these results will seem familiar or may have occurred to other researchers in the area of optimal control. The purpose of this paper is to make the distinction explicit and to establish the validity of these controllability theorems on their own merits. The theorems are demonstrated using a simple example and the principal result (a controllability minimum principle) is shown to be equivalent to the Kalman controllability criterion for linear systems.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Least-square methods ; variable-metric methods ; gradient methods ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract New algorithms are presented for approximating the minimum of the sum of squares ofM real and differentiable functions over anN-dimensional space. These algorithms update estimates for the location of a minimum after each one of the functions and its first derivatives are evaluated, in contrast with other least-square algorithms which evaluate allM functions and their derivatives at one point before using any of this information to make an update. These new algorithms give estimates which fluctuate about a minimum rather than converging to it. For many least-square problems, they give an adequate approximation for the solution more quickly than do other algorithms.
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  • 8
    ISSN: 1573-2878
    Keywords: Augmented penalty function ; method of multipliers ; penalty function methods ; nonlinear programming ; mathematical programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper describes an accelerated multiplier method for solving the general nonlinear programming problem. The algorithm poses a sequence of unconstrained optimization problems. The unconstrained problems are solved using a rank-one recursive algorithm described in an earlier paper. Multiplier estimates are obtained by minimizing the error in the Kuhn-Tucker conditions using a quadratic programming algorithm. The convergence of the sequence of unconstrained problems is accelerated by using a Newton-Raphson extrapolation process. The numerical effectiveness of the algorithm is demonstrated on a relatively large set of test problems.
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  • 9
    ISSN: 1573-2878
    Keywords: Aerospace engineering ; singular points ; nonlinear programming ; penalty-function methods ; variable-metric methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A heuristic method is presented for determining the equilibrium states of motion of dynamic systems, in particular, spacecraft. The method can also be applied to the solution of sets of linear or nonlinear algebraic equations. A positive-semidefinite functional is formed to convert the problem to that of finding those minimum points where the functional vanishes. The process is initiated within a selecteddomain of interest by random search; convergence to a minimum is obtained by a modified Davidon's deflected gradient technique. To render this approach feasible in the presence of constraints, the functional is modified to include penalty terms which cause the functional to approach infinity at the constraint boundaries. Close approximations to solutions near the constraint boundaries are found by applying Carroll's approach in successively reducing the weighting factors of the penalty terms. After finding a minimum, the local domain around this point is eliminated by adding to the functional an interior constraint term, representing the surface under a hypersphere centered at the minimum point. The domain of consideration now becomes the subdomain formed by subtracting the space contained within this hypersphere from the previous domain of interest. Minima are now sought within the remaining space, as before.
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  • 10
    ISSN: 1573-2878
    Keywords: Decomposition ; nonlinear programming ; structural optimization ; trusses
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A decomposition method, used in least-weight plastic design, is extended to solve problems with nonlinearity arising from variable structure geometry. The problem considered is that of finding vectorsx 1,x 2, andq that minimize [l max{|x 1|, |x 2|}], subject toAx 1=b 1 andAx 2=b 2, where both the vectorl and the matrixA are nonlinear functions ofq.
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