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  • Articles  (26)
  • Mathematics  (26)
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  • Articles  (26)
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  • 1
    Electronic Resource
    Electronic Resource
    Finite dimensional approximation in infinite dimensional mathematical programming (1992)
    Springer
    Mathematical programming 54 (1992), S. 307-333 
    Details
    ISSN: 1436-4646
    Keywords: Value convergence ; reachability ; solution set convergence ; tie-breaking ; stopping rule ; infinite horizon optimization ; production planning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the problem of approximating an optimal solution to a separable, doubly infinite mathematical program (P) with lower staircase structure by solutions to the programs (P(N)) obtained by truncating after the firstN variables andN constraints of (P). Viewing the surplus vector variable associated with theNth constraint as a state, and assuming that all feasible states are eventually reachable from any feasible state, we show that the efficient set of all solutions optimal to all possible feasible surplus states for (P(N)) converges to the set of optimal solutions to (P). A tie-breaking algorithm which selects a nearest-point efficient solution for (P(N)) is shown (for convex programs) to converge to an optimal solution to (P). A stopping rule is provided for discovering a value ofN sufficiently large to guarantee any prespecified level of accuracy. The theory is illustrated by an application to production planning.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01586057
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  • 2
    Electronic Resource
    Electronic Resource
    Pure adaptive search in global optimization (1992)
    Springer
    Mathematical programming 53 (1992), S. 323-338 
    Details
    ISSN: 1436-4646
    Keywords: Random search ; Monte Carlo optimization ; global optimization ; complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure adaptive seach iteratively constructs a sequence of interior points uniformly distributed within the corresponding sequence of nested improving regions of the feasible space. That is, at any iteration, the next point in the sequence is uniformly distributed over the region of feasible space containing all points that are strictly superior in value to the previous points in the sequence. The complexity of this algorithm is measured by the expected number of iterations required to achieve a given accuracy of solution. We show that for global mathematical programs satisfying the Lipschitz condition, its complexity increases at mostlinearly in the dimension of the problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585710
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  • 3
    Electronic Resource
    Electronic Resource
    Conditions for the discovery of solution horizons (1993)
    Springer
    Mathematical programming 59 (1993), S. 215-229 
    Details
    ISSN: 1436-4646
    Keywords: Dynamic programming ; deterministic and Markov ; infinite state: infinite stage problems ; programming ; infinite dimensional: infinite horizon optimization ; shortest paths ; infinite networks
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable. These problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network. We provide general forward algorithms with stopping rules for their solution. The key condition required is that of weak reachability, which roughly requires that for any sequence of nodes or states, it must be possible from optimal states to reach states close in cost to states along this sequence. Moreover the costs to reach these states must converge to zero. Applications are considered in optimal search, undiscounted Markov decision processes, and deterministic infinite horizon optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581244
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  • 4
    Electronic Resource
    Electronic Resource
    Random polytopes: Their definition, generation and aggregate properties (1982)
    Springer
    Mathematical programming 24 (1982), S. 39-54 
    Details
    ISSN: 1436-4646
    Keywords: Random Polytopes ; Linear Programming ; Problem Generation ; Aggregate Polytope Properties
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The definition of random polytope adopted in this paper restricts consideration to those probability measures satisfying two properties. First, the measure must induce an absolutely continuous distribution over the positions of the bounding hyperplanes of the random polytope; and second, it must result in every point in the space being equally as likely as any other point of lying within the random polytope. An efficient Monte Carlo method for their computer generation is presented together with analytical formulas characterizing their aggregate properties. In particular, it is shown that the expected number of extreme points for such random polytopes increases monotonically in the number of constraints to the limiting case of a polytope topologically equivalent to a hypercube. The implied upper bound of 2 n wheren is the dimensionality of the space is significantly less than McMullen's attainable bound on the maximal number of vertices even for a moderate number of constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585093
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  • 5
    Electronic Resource
    Electronic Resource
    The expected number of extreme points of a random linear program (1986)
    Springer
    Mathematical programming 35 (1986), S. 129-134 
    Details
    ISSN: 1436-4646
    Keywords: Random linear program ; random polytope ; extreme points
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract There has been increasing attention recently on average case algorithmic performance measures since worst case measures can be qualitatively quite different. An important characteristic of a linear program, relating to Simplex Method performance, is the number of vertices of the feasible region. We show 2 n to be an upper bound on the mean number of extreme points of a randomly generated feasible region with arbitrary probability distributions on the constraint matrix and right hand side vector. The only assumption made is that inequality directions are chosen independently in accordance with a series of independent fair coin tosses.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580643
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  • 6
    Electronic Resource
    Electronic Resource
    Pure adaptive search in monte carlo optimization (1989)
    Springer
    Mathematical programming 43 (1989), S. 317-328 
    Details
    ISSN: 1436-4646
    Keywords: Random search ; Monte Carlo optimization ; convex programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure adaptive search constructs a sequence of points uniformly distributed within a corresponding sequence of nested regions of the feasible space. At any stage, the next point in the sequence is chosen uniformly distributed over the region of feasible space containing all points that are equal or superior in value to the previous points in the sequence. We show that for convex programs the number of iterations required to achieve a given accuracy of solution increases at most linearly in the dimension of the problem. This compares to exponential growth in iterations required for pure random search.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582296
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  • 7
    Electronic Resource
    Electronic Resource
    Duality in infinite dimensional linear programming (1992)
    Springer
    Mathematical programming 53 (1992), S. 79-97 
    Details
    ISSN: 1436-4646
    Keywords: Infinite dimensional linear program ; duality ; infinite horizon optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585695
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  • 8
    Electronic Resource
    Electronic Resource
    Simulated annealing for constrained global optimization (1994)
    Springer
    Journal of global optimization 5 (1994), S. 101-126 
    Details
    ISSN: 1573-2916
    Keywords: Continuous simulated annealing ; adaptive cooling ; random search ; global optimization ; Monte Carlo optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Hide-and-Seek is a powerful yet simple and easily implemented continuous simulated annealing algorithm for finding the maximum of a continuous function over an arbitrary closed, bounded and full-dimensional body. The function may be nondifferentiable and the feasible region may be nonconvex or even disconnected. The algorithm begins with any feasible interior point. In each iteration it generates a candidate successor point by generating a uniformly distributed point along a direction chosen at random from the current iteration point. In contrast to the discrete case, a single step of this algorithm may generateany point in the feasible region as a candidate point. The candidate point is then accepted as the next iteration point according to the Metropolis criterion parametrized by anadaptive cooling schedule. Again in contrast to discrete simulated annealing, the sequence of iteration points converges in probability to a global optimum regardless of how rapidly the temperatures converge to zero. Empirical comparisons with other algorithms suggest competitive performance by Hide-and-Seek.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01100688
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  • 9
    Electronic Resource
    Electronic Resource
    Improving Hit-and-Run for global optimization (1993)
    Springer
    Journal of global optimization 3 (1993), S. 171-192 
    Details
    ISSN: 1573-2916
    Keywords: Random search ; Monte Carlo optimization ; algorithm complexity ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Improving Hit-and-Run is a random search algorithm for global optimization that at each iteration generates a candidate point for improvement that is uniformly distributed along a randomly chosen direction within the feasible region. The candidate point is accepted as the next iterate if it offers an improvement over the current iterate. We show that for positive definite quadratic programs, the expected number of function evaluations needed to arbitrarily well approximate the optimal solution is at most O(n5/2) wheren is the dimension of the problem. Improving Hit-and-Run when applied to global optimization problems can therefore be expected to converge polynomially fast as it approaches the global optimum.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096737
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  • 10
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    Parameter-Free Sampled Fictitious Play for Solving Deterministic Dynamic Programming Problems (2015)
    Springer
    Details
    Publication Date: 2015-08-25
    Print ISSN: 0022-3239
    Electronic ISSN: 1573-2878
    Topics: Mathematics
    Published by Springer
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