Accelerating two projection methods via perturbations with application to Intensity-Modulated Radiation Therapy

Constrained convex optimization problems arise naturally in many real-world applications. One strategy to solve them in an approximate way is to translate them into a sequence of convex feasibility problems via the recently developed level set scheme and then solve each feasibility problem using projection methods. However, if the problem is ill-conditioned, projection methods often show zigzagging behavior and therefore converge slowly. To address this issue, we exploit the bounded perturbation resilience of the projection methods and introduce two new perturbations which avoid zigzagging behavior. The first perturbation is in the spirit of $k$-step methods and uses gradient information from previous iterates. The second uses the approach of surrogate constraint methods combined with relaxed, averaged projections. We apply two different projection methods in the unperturbed version, as well as the two perturbed versions, to linear feasibility problems along with nonlinear optimization problems arising from intensity-modulated radiation therapy (IMRT) treatment planning. We demonstrate that for all the considered problems the perturbations can significantly accelerate the convergence of the projection methods and hence the overall procedure of the level set scheme. For the IMRT optimization problems the perturbed projection methods found an approximate solution up to 4 times faster than the unperturbed methods while at the same time achieving objective function values which were 0.5 to 5.1% lower.Comment: Accepted for publication in Applied Mathematics & Optimizatio

On the Asymptotic Average Number of Efficient Vertices in Multiple Objective Linear Programming

AbstractLeta1,…,am,c1,…,ckbe independent random points in Rnthat are identically distributed spherically symmetrical in Rnand letX≔{x∈Rn|aTix⩽1,i=1,…,m} be the associated random polyhedron form⩾n⩾2. We consider multiple objective linear programming problems maxx∈XcT1x, maxx∈XcT2x,…,maxx∈XcTkxwith 1⩽k⩽n. For distributions with algebraically decreasing tail in the unit ball, we investigate the asymptotic expected number of vertices in the efficient frontier ofXwith respect toc1,…,ckfor fixedn,kandm→∞. This expected number of efficient vertices is the most significant indicator for the average-case complexity of the multiple objective linear programming problem

An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence

Semi-infinite programming can be used to model a large variety of complex optimization problems. The simple description of such problems comes at a price: semi-infinite problems are often harder to solve than finite nonlinear problems. In this paper we combine a classical adaptive discretization method developed by Blankenship and Falk and techniques regarding a semi-infinite optimization problem as a bi-level optimization problem. We develop a new adaptive discretization method which combines the advantages of both techniques and exhibits a quadratic rate of convergence. We further show that a limit of the iterates is a stationary point, if the iterates are stationary points of the approximate problems

A generalized projection-based scheme for solving convex constrained optimization problems

In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility problems by iteratively constraining the objective function from above until the feasibility problem is inconsistent. For each of the feasibility problems one may apply any of the existing projection methods for solving it. In particular, the scheme allows the use of subgradient projections and does not require exact projections onto the constraints sets as in existing similar methods. We also apply the newly introduced concept of superiorization to optimization formulation and compare its performance to our scheme. We provide some numerical results for convex quadratic test problems as well as for real-life optimization problems coming from medical treatment planning.Comment: Accepted to publication in Computational Optimization and Application

Metronomic Treatment with Low-Dose Trofosfamide Leads to a Long-Term Remission in a Patient with Docetaxel-Refractory Advanced Metastatic Prostate Cancer

The treatment of metastatic prostate cancer patients refractory to androgen withdrawal and docetaxel therapy is currently discouraging and new therapeutic approaches are vastly needed. Here, we report a long-term remission over one year in a 68-year-old patient with metastatic docetaxel-refractory prostate cancer employing low-dose trofosfamide. The patient suffered from distant failure with several bone lesions and lymph node metastases depicted by a (11) C-Choline positron emission tomography/computerized tomography (PET/CT). After initiation of trofosfamide 100 mg taken orally once a day we observed a steadily decreasing PSA value from initial 46.6 down to 2.1 μg/L. The Choline-PET/CT was repeated after 10 months of continuous therapy and demonstrated a partial remission of the bone lesions and a regression of all involved lymph nodes but one. Taken together we found an astonishing and durable activity of the alkylating agent trofosfamide given in a metronomic fashion. We rate the side effects as low and state an excellent therapeutic ratio of this drug in our patient

Sensitivity analysis for lexicographic ordering in radiation therapy treatment planning

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/134937/1/mp0218.pd

Costlets: A Generalized Approach to Cost Functions for Automated Optimization of IMRT Treatment Plans

We present the creation and use of a generalized cost function methodology based on costlets for automated optimization for conformal and intensity modulated radiotherapy treatment plans. In our approach, cost functions are created by combining clinically relevant “costlets”. Each costlet is created by the user, using an “evaluator” of the plan or dose distribution which is incorporated into a function or “modifier” to create an individual costlet. Dose statistics, dose-volume points, biological model results, non-dosimetric parameters, and any other information can be converted into a costlet. A wide variety of different types of costlets can be used concurrently. Individual costlet changes affect not only the results for that structure, but also all the other structures in the plan (e.g., a change in a normal tissue costlet can have large effects on target volume results as well as the normal tissue). Effective cost functions can be created from combinations of dose-based costlets, dose-volume costlets, biological model costlets, and other parameters. Generalized cost functions based on costlets have been demonstrated, and show potential for allowing input of numerous clinical issues into the optimization process, thereby helping to achieve clinically useful optimized plans. In this paper, we describe and illustrate the use of the costlets in an automated planning system developed and used clinically at the University of Michigan Medical Center. We place particular emphasis on the flexibility of the system, and its ability to discover a variety of plans making various trade-offs between clinical goals of the treatment that may be difficult to meet simultaneously.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47484/1/11081_2005_Article_2066.pd

On the Variance of Additive Random Variables on Stochastic Polyhedra

Let $a_i i:= 1,\dots,m.$ be an i.i.d. sequence taking values in $\mathbb{R}^n$. Whose convex hull is interpreted as a stochastic polyhedron $P$. For a special class of random variables which decompose additively relative to their boundary simplices, eg. the volume of $P$, integral representations of their first two moments are given which lead to asymptotic estimations of variances for special "additive variables" known from stochastic approximation theory in case of rotationally symmetric distributions