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Properties of the LogBarrier Function on Degenerate Nonlinear Programs
 MATH. OPER. RES
, 1999
"... We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independ ..."
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We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independent. When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution, and obtain a semiexplicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution.
The role of linear objective functions in barrier methods: Corrigenda
 Mathematical Programming, Series A
, 2000
"... . The published paper contains a number of typographical errors and an incomplete proof. We indicate the corrections here. Our paper [1] contains the following typographical errors. Page 364, statement of Proposition 1. Replace "Assume that (30) is satisfied: : :" by "Assume that the ..."
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. The published paper contains a number of typographical errors and an incomplete proof. We indicate the corrections here. Our paper [1] contains the following typographical errors. Page 364, statement of Proposition 1. Replace "Assume that (30) is satisfied: : :" by "Assume that the conditions of Theorem 1 hold and that (30) is satisfied: : :". Equation (31). Replace the exponent "oe \Gamma 1" by "oe". Equation (37), second displayed line. Replace "o( 1+oe=2 )" by "O( 1+oe=2 )". Equation (47). Delete "i = q + 1; : : : ; m". Page 370, line 10. Replace "~oe ! oe" by "1 ~ oe ! oe 2". Similarly, on line 2 of Algorithm NL, replace "0 ! ~ oe ! oe" by "1 ~ oe ! oe 2". The final part of the proof of Theorem 2 is incomplete. We remedy this fault by deleting the material from line 6 on page 368 through the end of the proof, and replacing with the following. For the term in brackets, we have 1 +O( oe\Gamma1 ) (1 \Gamma )(=+ ) + +O( oe\Gamma1 )(=+ ) \Gamma 1 = \Gamma (1 \Gamma ...
A QQPMinimization Method for Semidefinite and Smooth Nonconvex Programs
 Working Paper, Abteilung Mathematik, Universitat
, 1998
"... . In several applications, semidefinite programs with nonlinear equality constraints arise. We give two such examples to emphasize the importance of this class of problems. We then propose a new solution method that also applies to smooth nonconvex programs. The method combines ideas of a predictor ..."
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. In several applications, semidefinite programs with nonlinear equality constraints arise. We give two such examples to emphasize the importance of this class of problems. We then propose a new solution method that also applies to smooth nonconvex programs. The method combines ideas of a predictor corrector interiorpoint method, of the SQP method, and of trust region methods. In particular, we believe that the new method combines the advantagesgenerality and robustness of trust region methods, local convergence of the SQPmethod and dataindependence of interiorpoint methods. Some convergence results are given, and some very preliminary numerical experiments suggest a high robustness of the proposed method. AMS 1991 subject classification. Primary: 90C. Key words. Predictor corrector method, SQP method, trust region method, semidefinite program. 1. Introduction This work was motivated by two applications from semidefinite programming with nonlinear equality constraints as outlin...
Level sets and non Gaussian integrals of positively homogeneous functions. arXiv:1110.6632v3
, 2011
"... Abstract. We investigate various properties of the sublevel set {x: g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions. For instance, the latter integral reduces to integrating h exp(−g) on the whole space Rn (a non Gaussian integral) and when ..."
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Abstract. We investigate various properties of the sublevel set {x: g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions. For instance, the latter integral reduces to integrating h exp(−g) on the whole space Rn (a non Gaussian integral) and when g is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of g. In fact, whenever h is nonnegative, the functional φ(g(x))h(x)dx is a convex function of g for a large class of functions φ: R+ → R. We also provide a numerical approximation scheme to compute the volume or integrate h (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set {x: g(x) ≤ 1} of minimum volume that contains some given subset K is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussianlike property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function. 1.
Exponential Barrier Method in Solving Line Programming Problems
 International Journal of Engineering & Technology IJETIJENS
, 2012
"... Abstract—This paper is concerned with the study of the exponential barrier method for linear programming problems with the essential property that each exponential barrier method is concave when viewed as a function of the multiplier.. It presents some background of the method and its variants for t ..."
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Abstract—This paper is concerned with the study of the exponential barrier method for linear programming problems with the essential property that each exponential barrier method is concave when viewed as a function of the multiplier.. It presents some background of the method and its variants for the problem. Under certain assumption on the parameters of the exponential barrier function, we give a rule for choosing the parameters of the barrier function. Theorems and algorithms for the methods are also given in this paper. At the end of the paper we give some conclusions and comments on the methods. Index Terms—linear programming, exponential barrier method, barrier function. Interior pint algorithm. I.
Exponential Penalty Methods for Solving Linear Programming Problems
 International Journal of Engineering Research and Applications
, 2012
"... This paper is concerned with the study of the exponential penalty method for linear programming problems with the essential property that each exponential penalty method is convex when viewed as a function of the multiplier.. It presents some background of the method and its variants for the problem ..."
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This paper is concerned with the study of the exponential penalty method for linear programming problems with the essential property that each exponential penalty method is convex when viewed as a function of the multiplier.. It presents some background of the method and its variants for the problem. Under certain assumption on the parameters of the exponential penalty function, we give a rule for choosing the parameters of the penalty function. Theorems and algorithms for the methods are also given in this paper. At the end of the paper we give some conclusions and comments on the methods. Key Words—exponential penalty method, linear programming problems, penalty parameter. I.
An Algorithm for Perturbed Secondorder Cone Programs
, 2004
"... The secondorder cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Qquadratically to a soluti ..."
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The secondorder cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Qquadratically to a solution of the perturbed problem. The algorithm is globalized. Numerical examples show that the algorithm is good for “warm starting” – for some instances, the solution of a perturbed problem is hit in two iterations.
The Behavior of Newtontype Methods on Two Equivalent Systems from Linear Programming
, 1998
"... Newtontype methods are fundamental techniques for solving systems of nonlinear equations. However, it is often not fully appreciated that these methods can produce significantly different behavior when applied to equivalent systems. In this paper, we investigate differences in local and global beha ..."
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Newtontype methods are fundamental techniques for solving systems of nonlinear equations. However, it is often not fully appreciated that these methods can produce significantly different behavior when applied to equivalent systems. In this paper, we investigate differences in local and global behavior of Newtontype methods when applied to two different but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier formulation, and the perturbed optimality conditions. Through theoretical analysis and numerical results, we show Newtontype methods perform more effectively on the latter system. 1 Introduction Newton's method is generally accepted as an effective tool for solving a system of nonlinear equations, F (v) = 0; where F : R n ! R n . It is a locally and quadratically convergent method under reasonable assumptions (see e.g. Dennis and Schnabel, [1]). In many practical applications, globally convergent methods are required to solve no...
Local Behavior of the Newton Method on Two Equivalent Systems from Linear Programming
"... Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce significantly different behavior when applied to equivalent systems ..."
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Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce significantly different behavior when applied to equivalent systems, i.e., problems with the same solution but different mathematical formulations. In this paper, we investigate differences in the local behavior of Newton's method when applied to two different but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier function formulation, and the equations in the socalled perturbed optimality conditions. Through theoretical analysis and numerical results, we provide an explanation of why Newton's method performs more effectively on the latter system.
Polynomial Barrier Method for Solving Linear Programming Problems
"... Abstract—In this work, we study a class of polynomial ordereven barrier functions for solving linear programming problems with the essential property that each member is concave polynomial ordereven when viewed as a function of the multiplier. Under certain assumption on the parameters of the barr ..."
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Abstract—In this work, we study a class of polynomial ordereven barrier functions for solving linear programming problems with the essential property that each member is concave polynomial ordereven when viewed as a function of the multiplier. Under certain assumption on the parameters of the barrier function, we give a rule for choosing the parameters of the barrier function. We also give an algorithm for solving this problem. Index Terms—linear programming, barrier method, polynomial ordereven. I.