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Title: Asumptotic behavior of trajectories associated with the exponential penalty in linear programming

Abstract

We consider the exponential penality function f(x, r) = c{prime} x + r {Sigma} exp[A{sub i}x - b{sub i}/r] associated with a linear program of the form min {l_brace}c{prime}x : Ax {<=} b{r_brace}. We show that for r close to 0, the unique unconstrained minimizer x(r) of f({center_dot}, r) admits an symptotic expansion of the form x(r) = x* + rd* + {eta}(r) where x* is a particular optimal solution of the linear program and the error term {eta}(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectory {Lambda}(r) which converges exponentially fast to a particular dual optimal solution. Then we study the asymptotic behavior of the solutions of the steepest descent differential equation u(t) = - {del}{sub x}f(u(t), r(t)), u(t{sub 0}) = u{sub 0}; showing that, under suitable conditions on the rate of decrease of r(t), u(t) converges towards an optimal solution {bar u} of the linear program. In particular, if r(t) decays slowly we find that {bar u} = x*.

Authors:
Publication Date:
OSTI Identifier:
35913
Report Number(s):
CONF-9408161-
TRN: 94:009753-0177
Resource Type:
Conference
Resource Relation:
Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
Country of Publication:
United States
Language:
English
Subject:
99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; DIFFERENTIAL EQUATIONS; ASYMPTOTIC SOLUTIONS; NUMERICAL SOLUTION; LINEAR PROGRAMMING

Citation Formats

Cominetti, R. Asumptotic behavior of trajectories associated with the exponential penalty in linear programming. United States: N. p., 1994. Web.
Cominetti, R. Asumptotic behavior of trajectories associated with the exponential penalty in linear programming. United States.
Cominetti, R. 1994. "Asumptotic behavior of trajectories associated with the exponential penalty in linear programming". United States.
@article{osti_35913,
title = {Asumptotic behavior of trajectories associated with the exponential penalty in linear programming},
author = {Cominetti, R},
abstractNote = {We consider the exponential penality function f(x, r) = c{prime} x + r {Sigma} exp[A{sub i}x - b{sub i}/r] associated with a linear program of the form min {l_brace}c{prime}x : Ax {<=} b{r_brace}. We show that for r close to 0, the unique unconstrained minimizer x(r) of f({center_dot}, r) admits an symptotic expansion of the form x(r) = x* + rd* + {eta}(r) where x* is a particular optimal solution of the linear program and the error term {eta}(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectory {Lambda}(r) which converges exponentially fast to a particular dual optimal solution. Then we study the asymptotic behavior of the solutions of the steepest descent differential equation u(t) = - {del}{sub x}f(u(t), r(t)), u(t{sub 0}) = u{sub 0}; showing that, under suitable conditions on the rate of decrease of r(t), u(t) converges towards an optimal solution {bar u} of the linear program. In particular, if r(t) decays slowly we find that {bar u} = x*.},
doi = {},
url = {https://www.osti.gov/biblio/35913}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Sat Dec 31 00:00:00 EST 1994},
month = {Sat Dec 31 00:00:00 EST 1994}
}

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