To closely describe the earliness/tardiness production planning problems in the JIT environment, a nonlinear semi-infinite programming model is proposed. Due to the issues of nonconvexivity and having infinitely many ...
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To closely describe the earliness/tardiness production planning problems in the JIT environment, a nonlinear semi-infinite programming model is proposed. Due to the issues of nonconvexivity and having infinitely many constraints, instead of applying traditional optimization approaches, a specially designed genetic algorithm with mutation along the negative gradient direction is developed. The proposed algorithm is a combination of the steepest descent method with the stochastic sampling algorithm. Some numerical results are included to show its potential for industrial applications.
By using the theory of parametric semi-infinite programming, we show that the solution of a linear semi-infinite programming problem can be obtained by solving a sequence of optimization problems with a single constra...
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By using the theory of parametric semi-infinite programming, we show that the solution of a linear semi-infinite programming problem can be obtained by solving a sequence of optimization problems with a single constraint.
We study the asymptotic behavior of interior-point methods for linear programming problems. Attempts to solve larger problems using interior-point methods lead to the question of how these algorithms behave as n (the ...
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We study the asymptotic behavior of interior-point methods for linear programming problems. Attempts to solve larger problems using interior-point methods lead to the question of how these algorithms behave as n (the number of variables) goes to infinity. Here, we take a different point of view and investigate what happens when n is infinite. Motivated by this approach, we study the limits of search directions, potential functions and central paths. We also suggest that the complexity of some linear programming problems may depend on the smoothness of the given problem rather than the number of variables. We prove that when n is infinite, for some subclasses of problems, one can still obtain a bound on the number of iterations required in terms of the smoothness of the problem and the desired accuracy.
This work examines the generalization of a certain interior-point method, namely the method of analytic centers, to semi-infinite linear programming problems. We define an analytic center for these problems and an app...
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This work examines the generalization of a certain interior-point method, namely the method of analytic centers, to semi-infinite linear programming problems. We define an analytic center for these problems and an appropriate norm to examine Newton's method for computing this center. A simple algorithm of order zero is constructed and a convergence proof for that algorithm is given. Finally, we describe a more practical implementation of a predictor-corrector method and give some numerical results. In particular we concentrate on practical integration rules that take care of the specific structure of the integrals.
Let Z be a compact set of the real space R with at least n + 2 points;f,h1,h2 : Z --> R continuous functions, h1,h2 strictly positive and P(x,z),x:=(x(0),...,x(n))(tau) is an element of Rn+1, z is an element of R, ...
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Let Z be a compact set of the real space R with at least n + 2 points;f,h1,h2 : Z --> R continuous functions, h1,h2 strictly positive and P(x,z),x:=(x(0),...,x(n))(tau) is an element of Rn+1, z is an element of R, a poly nomial of degree at most n. Consider a feasible set M := {x is an element of Rn+1\For All z is an element of Z, -h(2)(z)less than or equal to P(x,z)-f(z)less than or equal to h(1)(z)}. Here it is proved the null vector 0 of Rn+1 belongs to the compact convex hull of the gradients +/- (1,z,...,z(n)), where z is an element of Z are the index points in which the constraint functions are active for a given x* is an element of M, if and only if M is a singleton.
This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective f...
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This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems. These methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty. We give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
We present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. This is a technical condition that is always satisfied in t...
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We present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. This is a technical condition that is always satisfied in the case of semi-definite optimization. We derive Lipschitz and Holder expansions of approximate optimal solutions, under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem. We show how the theory applies to semi-infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds, and discuss the differentiability of metric projections as well as the Moreau-Yosida regularization. Finally we show how the theory applies to semi-definite optimization.
An error is pointed out in the local convergence proof in the quoted paper [J. L. Zhou and A. L. Tits, SIAM J. Optim., 6 (1996), pp. 461--487]. A correct proof is given.
An error is pointed out in the local convergence proof in the quoted paper [J. L. Zhou and A. L. Tits, SIAM J. Optim., 6 (1996), pp. 461--487]. A correct proof is given.
A cutting plane algorithm is proposed for estimating variance and covariance components by maximum likelihood or restricted maximum likelihood enforcing the constraints that covariance matrices be positive semidefinit...
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A cutting plane algorithm is proposed for estimating variance and covariance components by maximum likelihood or restricted maximum likelihood enforcing the constraints that covariance matrices be positive semidefinite. For tests of hypotheses involving these constrained estimates, an asymptotic parametric bootstrap is proposed for approximating the distribution of the likelihood ratio test statistic. Although-the bootstrap is generally inconsistent when the true parameter value is on the boundary of the feasible region, the double bootstrap can be used to show that the ordinary bootstrap works well in certain problems.
In this paper we study uniqueness of Lagrange multipliers in optimization problems subject to cone constraints. The main tool in our investigation of this question will be a calculus of dual (polar) cones. We give suf...
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In this paper we study uniqueness of Lagrange multipliers in optimization problems subject to cone constraints. The main tool in our investigation of this question will be a calculus of dual (polar) cones. We give sufficient and in some cases necessary conditions for uniqueness of Lagrange multipliers in general Banach spaces. General results are then applied to two particular examples of the semidefinite and semi-infinite programming problems, respectively.
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