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Discrete optimization problems (DOPs) arise in various applications such as planning, scheduling, computer aided design, robotics, game playing, and constraint directed reasoning. Often, a DOP is formulated in terms of finding a minimum cost solution path in a graph from an initial node to a goal node. It is solved using graph/tree search methods such as backtracking, branch-and-bound, heuristic search,...
In this paper we present two continuous-time models of economic competition which are based on a stochastic differential game formalism. We focus our presentation on the modeling possibilities offered by the frameworks of piecewise deterministic and switching diffusion control systems respectively. We develop two duopoly models: a dynamic R&D competition model and a stochastic fishery exploitation...
A survey of stability and sensitivity results for the solutions to parameter depenedent cone constrained optimization problems in abstract Banach spaces is presented. An application to optimal control problems for nonlinear ordinary differential equations subject to control and state constraints is given.
To minimize a convex function f, we state a penalty-type bundle algorithm, where the penalty uses a variable metric. This metric is updated according to quasi-Newton formulae based on Moreau-Yosida approximations of f. In particular, we introduce a “reversal” quasi-Newton formula, specially suited for our purpose. We consider several variants in the algorithm and discuss their respective merits. Furthermore,...
A new bundle method for minimizing a convex nondifferentiable function f:ℜ → ℜ is presented. At each iteration a master problem is solved to get a search direction d. This master problem is a quadratic programming problem of the type $$\begin{gathered}\min _d \tfrac{1}{2}d^T d \hfill \\s.t.\upsilon _c \geqslant g_i^T d - \varepsilon _i ,\forall i \in I \hfill \\\end{gathered}$$ where v ...
In this paper conjugate gradient methods with nonmonotone line search technique are introduced. This new line search technique is based on a relaxation of the strong Wolfe conditions and it allows to accept larger steps. The proposed conjugate gradient methods are still globally convergent and, at the same time, they should not suffer the propensity for short steps of some classical conjugate gradient...
A space transformation technique is used for the reduction of constrained minimization problems to minimization problems without inequality constraints. The continuous and discrete versions of Newton's method are applied for solving such reduced LP and NLP problems. The space transformation modifies these methods and introduces additional matrices which play the role of a multiplicative barrier, preventing...
We construct a generalization of affine-scaling vector fields for matrix linear programming problems. We discuss various properties of these vector fields and suggest a generalization of Dikin's algorithm.
We consider two kinds of nonconvex problems: convex maximization and reverseconvex optimization. Using the new information about the problems in the form of Global Optimality Search Algorithms [1–5], we construct Global Search Algorithms and study their global convergence. Numerical experiments also presented here are rather promising especially for large dimension problems.
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