Purpose: Intensity modulated proton therapy (IMPT) is sensitive to errors, mainly due to high density dependency and steep beam dose gradients. Conventional margins are often insufficient to ensure robustness of the plans. In this paper, 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 in 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, 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. *
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, 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 Karmarkar's 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.
Abstract. 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.
Abstract. Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization. Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton's method for a point satisfying the perturbed optimality conditions. These equations involve both the primal and dual variables and become increasingly ill-conditioned as the optimization proceeds. In this context, an iterative linear solver must not only handle the ill-conditioning but also detect the occurrence of KKT matrices with the wrong matrix inertia. A one-parameter family of equivalent linear equations is formulated that includes the KKT system as a special case. The discussion focuses on a particular system from this family, known as the "doubly augmented system," that is positive definite with respect to both the primal and dual variables. This property means that a standard preconditioned conjugate-gradient method involving both primal and dual variables will either terminate successfully or detect if the KKT matrix has the wrong inertia. Constraint preconditioning is a well-known technique for preconditioning the conjugate-gradient method on augmented systems. A family of constraint preconditioners is proposed that provably eliminates the inherent ill-conditioning in the augmented system. A considerable benefit of combining constraint preconditioning with the doubly augmented system is that the preconditioner need not be applied exactly. Two particular "active-set" constraint preconditioners are formulated that involve only a subset of the rows of the augmented system and thereby may be applied with considerably less work. Finally, some numerical experiments illustrate the numerical performance of the proposed preconditioners and highlight some theoretical properties of the preconditioned matrices.
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