Anders Forsgren

Interior Methods for Nonlinear Optimization

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, interior-point 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 Karmarkars widely publicized announcement in 1984 of a fast polynomial-time 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.

Minimax optimization for handling range and setup uncertainties in proton therapy.

PURPOSE Intensity modulated proton therapy (IMPT) is sensitive to errors, mainly due to high stopping power dependency and steep beam dose gradients. Conventional margins are often insufficient to ensure robustness of treatment plans. In this article, a method is developed that takes the uncertainties into account during the plan optimization. METHODS Dose contributions for a number of range and setup errors are calculated and a minimax optimization is performed. The minimax optimization aims at minimizing the penalty of the worst case scenario. Any optimization function from conventional treatment planning can be utilized by the method. By considering only scenarios that are physically realizable, the unnecessary conservativeness of other robust optimization methods is avoided. Minimax optimization is related to stochastic programming by the more general minimax stochastic programming formulation, which enables accounting for uncertainties in the probability distributions of the errors. RESULTS The minimax optimization method is applied to a lung case, a paraspinal case with titanium implants, and a prostate case. It is compared to conventional methods that use margins, single field uniform dose (SFUD), and material override (MO) to handle the uncertainties. For the lung case, the minimax method and the SFUD with MO method yield robust target coverage. The minimax method yields better sparing of the lung than the other methods. For the paraspinal case, the minimax method yields more robust target coverage and better sparing of the spinal cord than the other methods. For the prostate case, the minimax method and the SFUD method yield robust target coverage and the minimax method yields better sparing of the rectum than the other methods. CONCLUSIONS Minimax optimization provides robust target coverage without sacrificing the sparing of healthy tissues, even in the presence of low density lung tissue and high density titanium implants. Conventional methods using margins, SFUD, and MO do not utilize the full potential of IMPT and deliver unnecessarily high doses to healthy tissues.

Primal-Dual Interior Methods for Nonconvex Nonlinear Programming

This paper concerns large-scale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penalty-barrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primal-dual system similar to that proposed for interior methods. The augmented penalty-barrier function may be interpreted as a merit function for values of the primal and dual variables. An inertia-controlling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penalty-barrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.

Stability of Symmetric Ill-Conditioned Systems Arising in Interior Methods for Constrained Optimization

Many interior methods for constrained optimization obtain a search direction as the solution of a symmetric linear system that becomes increasingly ill-conditioned as the solution is approached. In some cases, this ill-conditioning is characterized by a subset of the diagonal elements becoming large in magnitude. It has been shown that in this situation the solution can be computed accurately regardless of the size of the diagonal elements. In this paper we discuss the formulation of several interior methods that use symmetric diagonally ill-conditioned systems. It is shown that diagonal ill-conditioning may be characterized by the property of strict

Newton methods for large-scale linear equality-constrained minimization

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Computing modified Newton directions using a partial Cholesky factorization

-diagonal dominance, which generalizes the idea of diagonal dominance to matrices whose diagonals are substantially larger in magnitude than the off-diagonals. A perturbation analysis is presented that characterizes the sensitivity of

Iterative Solution of Augmented Systems Arising in Interior Methods

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Inertia-controlling factorizations for optimization algorithms

-diagonally dominant systems under a certain class of structured perturbations. Finally, we give a rounding-error analysis of the symmetric indefinite factorization when applied to

Optimality conditions for nonconvex semidefinite programming

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On Linear Least-Squares Problems With Diagonally Dominant Weight Matrices

-diagonally dominant systems. This analysis resolves the (until now) open question of whether the class of perturbations used in the sensitivity analysis is representative of the rounding error made during the numerical solution of the barrier equations.