Robert R. Meyer

A variable-penalty alternating directions method for convex optimization

We study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints. The method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both Lagrangian and proximal terms. The minimizers determine the new proximal terms and a simple update of the Lagrangian terms follows. We prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers. Another novelty is that we allow penalty matrices, and these may vary per iteration. This can be beneficial in applications, since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties. As an application, we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems.

On the existence of optimal solutions to integer and mixed-integer programming problems

The purpose of this paper is to present sufficient conditions for the existence of optimal solutions to integer and mixed-integer programming problems in the absence of upper bounds on the integer variables. It is shown that (in addition to feasibility and boundedness of the objective function) (1) in the pure integer case a sufficient condition is that all of the constraints (other than non-negativity and integrality of the variables) beequalities, and (2) that in the mixed-integer caserationality of the constraint coefficients is sufficient. Some computational implications of these results are also given.

Moderate predictive value of demographic and behavioral characteristics for a diagnosis of HPV16-positive and HPV16-negative head and neck cancer

Patients with HPV-positive and HPV-negative head and neck squamous cell carcinoma (HNSCC) are significantly different with regard to sociodemographic and behavioral characteristics that clinicians may use to assume tumor HPV status. Machine learning methods were used to evaluate the predictive value of patient characteristics and laboratory biomarkers of HPV exposure for a diagnosis of HPV16-positive HNSCC compared to in situ hybridization, the current gold-standard. Models that used a combination of demographic characteristics such as age, tobacco use, gender, and race had only moderate predictive value for tumor HPV status among all patients with HNSCC (positive predictive value [PPV]=75%, negative predictive value [NPV]=68%) or when limited to oropharynx cancer patients (PPV=55%, NPV=65%) and thus included a sizeable number of false positive and false negative predictions. Prediction was not improved by the addition of other demographic or behavioral factors (sexual behavior, income, education) or biomarkers of HPV16 exposure (L1, E6/7 antibodies or DNA in oral exfoliated cells). Patient demographic and behavioral characteristics as well as HPV biomarkers are not an accurate substitute for clinical testing of tumor HPV status.

An Interior Point Method for Block Angular Optimization

An interior point method for block angular optimization is developed and the convergence properties of the method are described. A major motivation for such a method is that most of the computation is easily parallelized. Computational results are presented for a class of large-scale linear programming models. These models are multicommodity flow problems that arise from an Air Force (Military Airlift Command) application and generate problems as large as 100,000 rows and 300,000 columns.

Selection of beam orientations in intensity-modulated radiation therapy using single-beam indices and integer programming

While the process of IMRT planning involves optimization of the dose distribution, the procedure for selecting the beam inputs for this process continues to be largely trial-and-error. We have developed an integer programming (IP) optimization method to optimize beam orientation using mean organ-at-risk (MOD) data from single-beam plans. Two test cases were selected in which one organ-at-risk (OAR) and four OARs were simulated, respectively, along with a PTV. Beam orientation space was discretized in 10 degrees increments. For each beam orientation, a single-beam plan without intensity modulation and without constraints on OAR dose was generated and normalized to yield a mean PTV dose of 2 Gy and the corresponding MOD was calculated. The degree of OAR sparing was related to the average OAR MODs resulting from the beam orientations utilized with improvements of up to 10% at some dose levels. On the other hand, OAR DVHs in the IMRT plans were insensitive to beam numbers (in the 6-9 range) for similar average single-beam MODs. These MOD data were input to an IP optimization process, which then selected specified numbers of beam angles as inputs to a treatment planning system. Our results show that sets of beam angles with lower average single-beam MODs produce IMRT plans with better OAR sparing than manually selected beam angles. To optimize beam orientations, weights were assigned to each OAR following MOD input to the IP which was subsequently solved using the branch-and-cut algorithm. Seven-beam orientations obtained from solving the IP were applied to the test case with four OARs and the resulting plan with a dose prescription of 63 Gy was compared with an equi-spaced beam plan. The IP selected beams produced dose-volume improvements of up to 40% for OARs proximal to the PTV. Further improvement in the DVH can be obtained by increasing the weights assigned to these OARs but at the expense of the remaining OARs.

Parallel optimization for traffic assignment

Most large-scale optimization problems exhibit structures that allow the possibility of attack via algorithms that exhibit a high level of parallelism. The emphasis of this paper is the development of parallel optimization algorithms for a class of convex, block-structured problems. Computational experience is cited for some large-scale problems arising from traffic assignment applications. The algorithms considered here have the property that they allow such problems to be decomposed into a set of smaller optimization problems at each major iteration. These smaller problems correspond to linear single-commodity networks in the traffic assignment case, and they may be solved in parallel. Results are given for the distributed solution of such problems on the CRYSTAL multicomputer.

Radiation Treatment Planning: Mixed Integer Programming Formulations and Approaches

Radiation therapy is extensively used to treat a wide range of cancers. Due to the increasing complexities of delivery mechanisms, and the improved imaging devices that allow more accurate determination of cancer location, determination of high quality treatment plans via trial-and-error methods is impractical and computer optimization approaches to planning are becoming more critical and more difficult.

Alternating direction splittings for block angular parallel optimization

We develop and compare three decomposition algorithms derived from the method of alternating directions. They may be viewed as block Gauss-Seidel variants of augmented Lagrangian approaches that take advantage of block angular structure. From a parallel computation viewpoint, they are ideally suited to a data parallel environment. Numerical results for large-scale multicommodity flow problems are presented to demonstrate the effectiveness of these decomposition algorithmims on the Thinking Machines CM-5 parallel supercomputer relative to the widely-used serial optimization package MINOS 5.4.

Computational aspects of two-segment separable programming

Recursive separable programming algorithms based on local, two-segment approximations are described for the solution of separable convex programs. Details are also given for the computation of lower bounds on the optimal value by both a primal and a dual approach, and these approaches are compared. Computational comparisons of the methods are provided for a variety of test problems, including a water supply application (with more than 600 constraints and more than 900 variables) and an econometric modelling problem (with more than 200 variables).

A study of the development of a mexican petrochemical industry using mixed-integer programming☆

Abstract A large interactive system such as the petrochemical industry requires a model that can account for the different interactions among units, providing at the same time a suitable mathematical representation of the variables of interest. In this work, a model for the development of a Mexican petrochemical industry is presented. The system is formulated as a Mixed-Integer Programming model, where installing a process is compared on an economic basis to importing its corresponding product. This formulation lets the model take economies-of-scale into account which are shown to be a very decisive factor in the selection of chemical processes, since a simple linear model does not seem to provide an adequate tool for the planning and development of a Mexican petrochemical industry. A heuristic approach using multiple linear programs is also discussed.