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37
A trust region method based on interior point techniques for nonlinear programming
 Mathematical Programming
, 1996
"... Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direc ..."
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Cited by 113 (19 self)
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Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primaldual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, SQP iteration, barrier method, trust region method.
Interior methods for nonlinear optimization
 SIAM Review
, 2002
"... Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their ..."
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Cited by 89 (4 self)
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Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomialtime interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.
On the solution of equality constrained quadratic programming problems arising . . .
, 1998
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Analysis of Inexact TrustRegion SQP Algorithms
 RICE UNIVERSITY, DEPARTMENT OF
, 2000
"... In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximatio ..."
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Cited by 21 (2 self)
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In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximations of firstorder derivatives. Accuracy requirements in our trustregion SQP methods are adjusted based on feasibility and optimality of the iterates. Our accuracy requirements are stated in general terms, but we show how they can be enforced using information that is already available in matrixfree implementations of SQP methods. In the absence of inexactness our global convergence theory is equal to that of Dennis, ElAlem, Maciel (SIAM J. Optim., 7 (1997), pp. 177207). If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trustregion methods with inexact gradient information fo...
Second Order Methods For Optimal Control Of TimeDependent Fluid Flow
, 1999
"... Second order methods for open loop optimal control problems governed by the twodimensional instationary NavierStokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasiNewton methods as well as various variants o ..."
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Cited by 17 (5 self)
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Second order methods for open loop optimal control problems governed by the twodimensional instationary NavierStokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasiNewton methods as well as various variants of SQPmethods are developed for applications to optimal ow control and their complexity in terms of system solves is discussed. Local convergence and rate of convergence are proved. A numerical example illustrates the feasibility of solving optimal control problems for twodimensional instationary NavierStokes equations by second order numerical methods in a standard workstation environment. Previously such problems were solved by gradient type methods.
Superlinear Convergence of AffineScaling InteriorPoint Newton Methods for InfiniteDimensional Nonlinear Problems with Pointwise Bounds
, 1999
"... We develop and analyze a superlinearly convergent affinescaling interiorpoint Newton method for infinitedimensional problems with pointwise bounds in L p space. The problem formulation is motivated by optimal control problems with L p controls and pointwise control constraints. The finite ..."
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Cited by 14 (6 self)
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We develop and analyze a superlinearly convergent affinescaling interiorpoint Newton method for infinitedimensional problems with pointwise bounds in L p space. The problem formulation is motivated by optimal control problems with L p controls and pointwise control constraints. The finitedimensional convergence theory by Coleman and Li (SIAM J. Optim., 6 (1996), pp. 418445) makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity. Since these features are not available in our infinitedimensional framework, algorithmic changes are necessary to ensure fast local convergence. The main building block is a Newtonlike iteration for an affinescaling formulation of the KKTcondition. We demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution. Using a pointwise projection instead ...
A Convergent Infeasible InteriorPoint TrustRegion Method For Constrained Minimization
 SIAM Journal on Optimization
, 1999
"... We study an infeasible interiorpoint trustregion method for constrained minimization. This method uses a logarithmicbarrier function for the slack variables and updates the slack variables using secondorder correction. We show that if a certain set containing the iterates is bounded and the orig ..."
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Cited by 11 (0 self)
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We study an infeasible interiorpoint trustregion method for constrained minimization. This method uses a logarithmicbarrier function for the slack variables and updates the slack variables using secondorder correction. We show that if a certain set containing the iterates is bounded and the origin is not in the convex hull of the nearly active constraint gradients everywhere on this set, then any cluster point of the iterates is a 1storder stationary point. If the cluster point satisfies an additional assumption (which holds when the constraints are linear or when the cluster point satisfies strict complementarity and a local error bound holds), then it is a 2ndorder stationary point. Key words. Nonlinear program, logarithmicbarrier function, interiorpoint method, trustregion strategy, 1st and 2ndorder stationary points, semidefinite programming. 1 Introduction We consider the nonlinear program with inequality constraints: minimize f(x) subject to g(x) = [g 1 (x) g m (...
Analysis of Inexact TrustRegion InteriorPoint SQP Algorithms
, 1995
"... In this paper we analyze inexact trustregion interiorpoint (TRIP) sequential quadratic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applicati ..."
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Cited by 11 (7 self)
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In this paper we analyze inexact trustregion interiorpoint (TRIP) sequential quadratic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonlinear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is expensive. Often, the solution of linear systems and derivatives are computed inexactly yielding nonzero residuals. This paper analyzes the effect of the inexactness onto the convergence of TRIP SQP and gives practical rules to control the size of the residuals of these inexact calculations. It is shown that if the size of the residuals is of the order of both the size of the constraints and the trustregion radius, t...
Nonmonotone Trust Region Methods for Nonlinear Equality Constrained Optimization without a Penalty Function
 MATH. PROGRAM., SER. B
, 2000
"... We propose and analyze a class of penaltyfunctionfree nonmonotone trustregion methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint viol ..."
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Cited by 10 (6 self)
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We propose and analyze a class of penaltyfunctionfree nonmonotone trustregion methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the ByrdOmojokun class of algorithms, each step is composed of a quasinormal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trustregion subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filterbased methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.
A PrimalDual Active Set Algorithm For Bilaterally Control Constrained Optimal Control Problems
, 1998
"... A generalized MoreauYosida based primaldual active set algorithm for the solution of a representative class of bilaterally control constrained optimal control problems with boundary control is developed. The use of the generalized MoreauYosida approximation allows an efficient identification of t ..."
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Cited by 8 (2 self)
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A generalized MoreauYosida based primaldual active set algorithm for the solution of a representative class of bilaterally control constrained optimal control problems with boundary control is developed. The use of the generalized MoreauYosida approximation allows an efficient identification of the active and inactive sets at each iteration level. The method requires no stepsize strategy and exhibits a finite termination property for the discretized problem class. In a series of numerical tests the efficiency of the new algorithm is emphasized.