arXiv subjects

Anders Forsgren

Publications and source records attributed to Anders Forsgren.

At least 19 recordsLinked to original sources

arXiv2025

On the asymptotic behavior of a higher-order extrapolation primal-dual interior-point method for nonlinear programming

A trajectory-following primal--dual interior-point method solves nonlinear optimization problems with inequality and equality constraints by approximately finding points satisfying perturbed Karush--Kuhn--Tucker optimality conditions for a decreasing order of perturbation controlled by the barrier parameter. Under some conditions, there is a unique local correspondence between small residuals of the optimality conditions and points yielding that residual, and the solution on the barrier trajectory for the next barrier parameter can be approximated using an approximate solution for the current parameter. A framework using higher-order derivative information of the correspondence is analyzed in which an extrapolation step to the trajectory is first taken after each decrease of the barrier parameter upon reaching a sufficient approximation. It suffices asymptotically to only take extrapolation steps for convergence at the rate the barrier parameter decreases with when using derivative information of high enough order. Numerical results for quadratic programming problems are presented using extrapolation as accelerator.

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arXiv2025

Dosimetric impact of real-time re-optimization of proton pencil-beam scanning for moving targets

When treating moving tumors, the precise delivery of proton therapy by pencil beam scanning (PBS) is challenged by the interplay effect. Although several 4D-optimization methods have been proposed, what is the most beneficial motion management technique is still an open question. In this study, we wish to investigate the dosimetric impact of re-optimizing the PBS spot weights during the treatment delivery in response to, and in anticipation of, variations in the patient's breathing pattern. We simulate for PBS the implementation of a real-time adaptive framework based on principles from receding horizon control. We consider the patient motion as characterized by a one-dimensional amplitude signal and a 4DCT, to simulate breathing of variable frequency. The framework tracks the signal and predicts the future motion with uncertainty increasing with the length of the prediction horizon. After each delivered energy layer, the framework re-optimizes the spot weights of the next layer based on the delivered dose and the predicted motion. For three lung patients, we generate 500 variable breathing patterns to evaluate the dosimetric results of the framework and compare them to those of implementations of previously proposed non-adaptive methods. Compared to the best non-adaptive method, the adaptive framework improves the CTV D98 in the near-worst breathing scenario (5th percentile), from 96.4 to 98.9 % of the prescribed dose and considerably reduces the variation as measured by a mean decrease in the inter-quartile range by more than 80 %. The target coverage improvements are achieved without generally compromising target dose homogeneity or OAR dose. The study indicates that a motion-adaptive approach based on re-optimization of spot weights during delivery has the potential to substantially improve the dosimetric performance of PBS given fast and accurate models of patient motion.

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arXiv2024

Interplay-robust optimization for treating irregularly breathing lung patients with pencil beam scanning

The steep dose gradients obtained with pencil beam scanning allow for precise tumor targeting at the cost of high sensitivity to uncertainties. Robust optimization is commonly applied to mitigate uncertainties in density and patient setup, while its application to motion management, called 4D-robust optimization (4DRO), is typically accompanied by other motion mitigation techniques. In particular, current commercial implementations of 4DRO do not model the interplay effect between the delivery time structure and the patient's motion. Previously, it has been shown that Interplay-robust optimization (IPRO) can mitigate the interplay effect given uncertainty in the patient's breathing frequency. In this study, we investigate and evaluate IPRO in the context where the motion uncertainty is extended to also include variations in breathing amplitude. We model the patients' motion using synthetic 4DCTs, each created by deforming a reference CT based on a motion pattern obtained with 4DMRI. Each synthetic 4DCT contains multiple breathing cycles, partitioned into two sets for scenario generation: one for optimization and one for evaluation. Motion scenarios are then created by randomly concatenating breathing cycles varying in period and amplitude. In addition, a method considering a single breathing cycle for generating optimization scenarios (IPRO-1C) is developed to investigate to which extent robustness can be achieved with limited information. IPRO and IPRO-1C increased the target coverage for all patient cases in terms of the near-worst-case (5th percentile) CTV D98, compared to 4DRO. After normalization of plan doses to equal target coverage, IPRO with 49 scenarios resulted in the greatest decreases in OAR dose, with near-worst-case (95th percentile) improvements averaging 4.2 %. IPRO-1C with 9 scenarios, with comparable computational demands as 4DRO, decreased OAR dose by 1.7 %.

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arXiv2024

Finite Newton-Step Generation via Subspace Quasi-Newton Methods for Quadratic Problems

We study the curvature information sufficient for finite Newton-step generation by quasi-Newton methods on strictly convex quadratic problems. In exact arithmetic, the memoryless BFGS quasi-Newton method terminates finitely on such a problem when exact line search is used, although its quasi-Newton approximation incorporates curvature information associated with only one direction. We show that exact line search can be replaced by curvature information associated with one additional direction. Information associated with at most two directions suffices to generate the Newton step after finitely many iterations, independently of the preceding step lengths. More precisely, knowledge of the action of the Hessian on a suitably chosen subspace of dimension at most two is sufficient to generate a step that decomposes into a subspace Newton step and a component parallel to the next conjugate direction. Applying this construction recursively generates the full Newton step after finitely many iterations, after which a unit step reaches the minimizer. The required Hessian actions can also be recovered from gradient differences, yielding a first-order formulation. The resulting algorithm is intended as a constructive device for identifying sufficient curvature information, rather than as a practical alternative to the method of conjugate gradients.

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arXiv2023

A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming

The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions. We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.

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arXiv2022

Implications of Clinical Target Distribution Weighted Radiotherapy Optimization

Delineating and planning with respect to regions suspected to contain microscopic tumor cells is an inherently uncertain task in radiotherapy. The recently proposed \textit{clinical target distribution} (CTD) is an alternative to the conventional \textit{clinical target volume} (CTV), with initial promise. Previously, using the CTD in planning has primarily been evaluated in comparison to a conventionally defined CTV. We propose to compare the CTD approach against CTV margins of various sizes, dependent on the threshold at which the tumor infiltration probability is considered relevant. First, a theoretical framework is presented, concerned with optimizing the trade-off between the probability of sufficient target coverage and the penalties associated with high dose. From this framework we derive conventional CTV-based planning and contrast it with the CTD approach. The approaches are contextualized further by comparison with established methods for managing geometric uncertainties. Second, for both a one- and a three-dimensional phantom, we compare a set of CTD plans created by varying the target objective function weight against a set of plans created by varying both the target weight and the CTV margin size. The results show that CTD-based planning gives slightly inefficient trade-offs between the evaluation criteria for a case in which near-minimum target dose is the highest priority. However, in a case when sparing a proximal organ at risk is critical, the CTD is better at maintaining sufficiently high dose toward the center of the target. We conclude that CTD-based planning is a computationally efficient method for planning with respect to delineation uncertainties, but that the inevitable effects on the dose distribution should not be disregarded.

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arXiv2021

Quasi-Newton methods for minimizing a quadratic function subject to uncertainty

We investigate quasi-Newton methods for minimizing a strictly convex quadratic function which is subject to errors in the evaluation of the gradients. The methods all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. A BFGS quasi-Newton method is empirically known to behave very well on a quadratic problem subject to small errors. We also investigate large-error scenarios, in which the expected behavior is not so clear. In particular, we are interested in the behavior of quasi-Newton matrices that differ from the identity by a low-rank matrix, such as a memoryless BFGS method. Our numerical results indicate that for large errors, a memory-less quasi-Newton method often outperforms a BFGS method. We also consider a more advanced model for generating search directions, based on solving a chance-constrained optimization problem. Our results indicate that such a model often gives a slight advantage in final accuracy, although the computational cost is significantly higher.

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arXiv2021

An optimization derivation of the method of conjugate gradients

We give a derivation of the method of conjugate gradients based on the requirement that each iterate minimizes a strictly convex quadratic on the space spanned by the previously observed gradients. Rather than verifying that the search direction has the correct properties, we show that generation of such iterates is equivalent to generation of orthogonal gradients which gives the description of the direction and the step length. Our approach gives a straightforward way to see that the search direction of the method of conjugate gradients is a negative scalar times the gradient of minimum Euclidean norm evaluated on the affine span of the iterates generated so far.

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arXiv2020

Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

The focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton's method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

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arXiv2020

Exact linesearch limited-memory quasi-Newton methods for minimizing a quadratic function

The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the method of preconditioned conjugate gradients, and hence give finite termination on quadratic optimization problems in exact arithmetic. With the framework of reduced-Hessians this class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. We give an indication of the performance of the methods within this framework by showing numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

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arXiv2019

On the existence of a short pivoting sequence for a linear program

Pivoting methods are of vital importance for linear programming, the simplex method being the by far most well-known. In this paper, a primal-dual pair of linear programs in canonical form is considered. We show that there exists a sequence of pivots, whose length is bounded by the minimum dimension of the constraint matrix, such that the pivot creates a nonsingular submatrix of the constraint matrix which increases by one row and one column at each iteration. Solving a pair of linear equations for each of these submatrices generates a sequence of optimal solutions of a primal-dual pair of linear programs of increasing dimensions, originating at the origin. The optimal solutions to the original primal-dual pair of linear programs are obtained in the final step. It is only an existence result, we have not been able to generate any rules based on properties of the problem to generate the sequence. The result is obtained by a decomposition of the final basis matrix.

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arXiv2018

On the Interplay between Robustness and Dynamic Planning for Adaptive Radiation Therapy

Interfractional geometric uncertainties can lead to deviations of the actual delivered dose from the prescribed dose distribution. To better handle these uncertainties during treatment, the authors propose a dynamic framework for robust adaptive radiation therapy in which a variety of robust adaptive treatment strategies are introduced and evaluated. This variety is a result of optimization variables with various degrees of freedom within robust optimization models that vary in their grade of conservativeness. We formulate an adaptive framework with dependence on time and/or uncertainty or neither. The framework is evaluated with either expected-value-, worst-case-, or conditional-value-at-risk-(CVaR)-optimization. The goal of this study is to identify the mathematical properties of the proposed robust adaptive strategies in order to restore the planned dose distribution. We apply a result from convex analysis to show that in our framework, the time-dependent problem has a time-independent optimal solution. This implies that the time-dependent problem can be solved by dynamically solving the corresponding time-independent problem. This is especially relevant in a clinical context, since it implies that adaptive radiation therapy can be implemented in clinical practice using commercially available treatment planning systems. We demonstrate that time-and-scenario-dependent variables are most compatible with worst-case-optimization and that including dose prediction potentially improves treatment compared to non-adaptive planning. Contrarily, time-and-scenario-independent variables appear most compatible with expected-value-optimization. Non-adaptive robust strategies may provide sufficient target coverage, comparable to robust adaptive strategies, if the occurring uncertainties follow the same distribution as those included in the robust model.

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arXiv2018

Increased accuracy of planning tools for optimization of dynamic multileaf collimator delivery of radiotherapy through reformulated objective functions

The purpose of this study is to examine in a clinical setting a novel formulation of objective functions for intensity-modulated radiotherapy treatment plan multicriteria optimization (MCO) that we suggested in a recent study. The proposed objective functions are extended with dynamic multileaf collimator (DMLC) delivery constraints from the literature, and a tailored interior point method is described to efficiently solve the resulting optimization formulation. In a numerical planning study involving three patient cases, DMLC plans Pareto optimal to the MCO formulation with the proposed objective functions are generated. Evaluated based on pre-defined plan quality indices, these DMLC plans are compared to conventionally generated DMLC plans. Comparable or superior plan quality is observed. Supported by these results, the proposed objective functions are argued to have a potential to streamline the planning process, since they are designed to overcome the methodological shortcomings associated with the conventional penalty-based objective functions assumed to cause the current need for time-consuming trial-and-error parameter tuning. In particular, the increased accuracy of the planning tools imposed by the proposed objective functions has the potential to make the planning process less complicated. These conclusions position the proposed formulation as an alternative to existing methods for automated planning.

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arXiv2018

On tradeoffs between treatment time and plan quality of volumetric-modulated arc therapy with sliding-window delivery

The purpose of this study is to give an exact formulation of optimization of volumetric-modulated arc therapy (VMAT) with sliding-window delivery, and to investigate the plan quality effects of decreasing the number of sliding-window sweeps made on the 360-degree arc for a faster VMAT treatment. In light of the exact formulation, we interpret an algorithm previously suggested in the literature as a heuristic method for solving this optimization problem. By first making a generalization, we suggest a modification of this algorithm for better handling of plans with fewer sweeps. In a numerical study involving one prostate and one lung case, plans with varying treatment times and number of sweeps are generated. It is observed that, as the treatment time restrictions become tighter, fewer sweeps may lead to better plan quality. Performance of the original and the modified version of the algorithm is evaluated in parallel. Applying the modified version results in better objective function values and less dose discrepancies between optimized and accurate dose, and the advantages are pronounced with decreasing number of sweeps.

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arXiv2017

On exact linesearch quasi-Newton methods for minimizing a quadratic function

This paper concerns exact linesearch quasi-Newton methods for minimizing a quadratic function whose Hessian is positive definite. We show that by interpreting the method of conjugate gradients as a particular exact linesearch quasi-Newton method, necessary and sufficient conditions can be given for an exact linesearch quasi-Newton method to generate a search direction which is parallel to that of the method of conjugate gradients. We also analyze update matrices and give a complete description of the rank-one update matrices that give search direction parallel to those of the method of conjugate gradients. In particular, we characterize the family of such symmetric rank-one update matrices that preserve positive definiteness of the quasi-Newton matrix. This is in contrast to the classical symmetric-rank-one update where there is no freedom in choosing the matrix, and positive definiteness cannot be preserved. The analysis is extended to search directions that are parallel to those of the preconditioned method of conjugate gradients in a straightforward manner.

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arXiv2017

Explicit optimization of plan quality measures in intensity-modulated radiation therapy treatment planning

Conventional planning objectives in optimization of intensity-modulated radiotherapy treatment (IMRT) plans are designed to minimize the violation of dose-volume histogram (DVH) thresholds using penalty functions. Although successful in guiding the DVH curve towards these thresholds, conventional planning objectives offer limited control of the individual points on the DVH curve (doses-at-volume) used to evaluate plan quality. In this study, we abandon the usual penalty-function framework and propose planning objectives that more explicitly relate to DVH statistics. The proposed planning objectives are based on mean-tail-dose, resulting in convex optimization. We also demonstrate how to adapt a standard optimization method to the proposed formulation in order to obtain a substantial reduction in computational cost. We investigate the potential of the proposed planning objectives as tools for optimizing DVH statistics through juxtaposition with the conventional planning objectives on two patient cases. Sets of treatment plans with differently balanced planning objectives are generated using either the proposed or the conventional approach. Dominance in the sense of better distributed doses-at-volume is observed in plans optimized within the proposed framework, indicating that the DVH statistics are better optimized and more efficiently balanced using the proposed planning objectives.

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arXiv2016

Towards Robust Adaptive Radiation Therapy Strategies

The authors propose robust adaptive strategies based on stochastic minimax optimization for a series of simulated treatments on a one-dimensional patient phantom. The plan applied during the first fractions should be able to handle anticipated systematic and random errors. At scheduled fractions, the impact of the measured errors on the delivered dose distribution is evaluated. For a patient receiving a dose that does not satisfy clinical goals, the plan is reoptimized based on these individually measured errors. The adapted plan is then applied during subsequent fractions until a new scheduled adaptation becomes necessary. In the first adaptive strategy, the measured systematic and random error scenarios and their assigned probabilities are updated to guide the robust reoptimization. In the second strategy, the grade of conservativeness is adapted in response to the measured dose delivery errors. In the third strategy, the uncertainty margins around the target are recalculated based on the measured errors. The simulated systematic and random errors are either similar to the anticipated errors or unpredictably larger, in order to critically evaluate the performance of the adaptive strategies. The simulations show that robustly optimized treatment plans provide sufficient treatment quality for those treatment error scenarios similar to the anticipated error scenarios. Combining robust planning with adaptation leads to improved OAR protection. In case of unpredictably larger treatment errors, the first strategy performs best at improving treatment quality in terms of target coverage and OAR protection. According to the simulations, these robust adaptive treatment strategies are able to identify candidates for an adaptive treatment, thus giving the opportunity to provide individualized plans, and improve their treatment quality through adaptation.

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arXiv2015

Primal and dual active-set methods for convex quadratic programming

Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primal-dual form. The primal method maintains feasibility of the primal inequalities while driving the infeasibilities of the dual inequalities to zero. The dual method maintains feasibility of the dual inequalities while moving to satisfy the primal inequalities. In each of these methods, the search directions satisfy a KKT system of equations formed from Hessian and constraint components associated with an appropriate column basis. The composition of the basis is specified by an active-set strategy that guarantees the nonsingularity of each set of KKT equations. Each of the proposed methods is a conventional active-set method in the sense that an initial primal- or dual-feasible point is required. In the second part of the paper, it is shown how the quadratic program may be solved as a coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simple-bound constraints and adding a penalty term to the objective function. Any conventional column basis may be made optimal for such a primal-dual pair of shifted-penalized problems. The shifts are then updated using the solution of either the primal or the dual shifted problem. An obvious application of this approach is to solve a shifted dual QP to define an initial feasible point for the primal (or vice versa). The computational performance of each of the proposed methods is evaluated on a set of convex problems.

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