Iterative Solution Methods for Beam Angle and Fluence Map Optimization in Intensity Modulated Radiation Therapy Planning

We present computational approaches for optimizing beam angles and fluence maps in Intensity Modulated Radiation Therapy (IMRT) planning. We assume that the number of angles to be used for the treatment is given by the treatment planner. A mixed integer programming (MIP) model and a linear programming (LP) model are used to find an optimal … Read more

A pricing problem under Monge property

We study a pricing problem where buyers with non-uniform demand purchase one of many items. Each buyer has a known benefit for each item and purchases the item that gives the largest utility, which is defined to be the difference between the benefit and the price of the item. The optimization problem is to decide … Read more

An efficient conjugate direction method with orthogonalization for large-scale quadratic optimization problems

A new conjugate direction method is proposed, which is based on an orthogonalization procedure and does not make use of line searches for the conjugate vector set construction. This procedure prevents the set of conjugate vectors from degeneracy and eliminates high sensitivity to computation errors pertinent to methods of conjugate directions, resulting in an efficient … Read more

An Extension of the Conjugate Directions Method With Orthogonalization to Large-Scale Problems With Bound Constraints

In our reports on GAMM-04 and ECCOMAS-04 there has been presented a new conjugate directions method for large scale unconstrained minimization problems. High efficiency of this method is ensured by employing an orthogonalization procedure: when constructing the next conjugate vector the component of the gradient is used that is orthogonal to the subspace of preceding … Read more

An Optimization Approach to Computing the Implied Volatility of American Options

We present a method to compute the implied volatility of American options as a mathematical program with equilibrium constraints. The formulation we present is new, as are the convergence results we prove. The algorithm holds the promise of being practical to implement, and we demonstrate some preliminary numerical results to this end. Citation Princeton University … Read more

Continuous optimization of beamlet intensities for photon and proton radiotherapy

Inverse approaches and, in particular, intensity modulated radiotherapy (IMRT), in combination with the development of new technologies such as multi-leaf collimators (MLCs), have enabled new potentialities of radiotherapy for cancer treatment. The main mathematical tool needed in this connection is numerical optimization. In this article, the variety of continuous optimization approaches, which have been proposed … Read more

SDPARA : SemiDefinite Programming Algorithm PARAllel Version

Abstract: The SDPA (SemiDefinite Programming Algorithm) is known as efficient computer software based on primal-dual interior-point method for solving SDPs (Semidefinite Programs). In many applications, however, some SDPs become larger and larger, too large for the SDPA to solve on a single processor. In execution of the SDPA applied to large scale SDPs, the computation … Read more

Computing Mountain Passes

We propose the elastic string algorithm for computing mountain passes in finite-dimensional problems. We analyze the convergence properties and numerical performance of this algorithm for benchmark problems in chemistry and discretizations of infinite-dimensional variational problems. We show that any limit point of the elastic string algorithm is a path that crosses a critical point at … Read more

Implementation and Evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0

The SDPA (SemiDefinite Programming Algorithm) is a software package for solving general SDPs(SemiDefinite Programs). It is written in C++ with the help of {\it LAPACK} for numerical linear algebra for dense matrix computation. The purpose of this paper is to present a brief description of the latest version of the SDPA and its high performance … Read more