Michael Kupferschmid

An ellipsoid algorithm for nonlinear programming

We investigate an ellipsoid algorithm for nonlinear programming. After describing the basic steps of the algorithm, we discuss its computer implementation and present a method for measuring computational efficiency. The computational results obtained from experimenting with the algorithm are discussed and the algorithms performance is compared with that of a widely used commercial code.

Solving nonlinear principal-agent problems using bilevel programming

While significant progress has been made, analytic research on principal-agent problems that seek closed-form solutions faces limitations due to tractability issues that arise because of the mathematical complexity of the problem. The principal must maximize expected utility subject to the agent’s participation and incentive compatibility constraints. Linearity of performance measures is often assumed and the Linear, Exponential, Normal (LEN) model is often used to deal with this complexity. These assumptions may be too restrictive for researchers to explore the variety of relationships between compensation contracts offered by the principal and the effort of the agent. In this paper we show how to numerically solve principal-agent problems with nonlinear contracts. In our procedure, we deal directly with the agent’s incentive compatibility constraint. We illustrate our solution procedure with numerical examples and use optimization methods to make the problem tractable without using the simplifying assumptions of a LEN model. We also show that using linear contracts to approximate nonlinear contracts leads to solutions that are far from the optimal solutions obtained using nonlinear contracts. A principal-agent problem is a special instance of a bilevel nonlinear programming problem. We show how to solve principal-agent problems by solving bilevel programming problems using the ellipsoid algorithm. The approach we present can give researchers new insights into the relationships between nonlinear compensation schemes and employee effort.

Computing the internal transient voltage response of a transformer with a nonlinear core using Gear's method. Part 1: Theory

An EHV transformers insulation structure must be designed to withstand the internal stresses generated during transients. Computer models are employed for predetermination of these stresses. This paper develops a detailed transformer model and solution method which represent the nonlinear, saturable characteristic of the magnetic core during transients. The resulting set of stiff nonlinear differential algebraic equations are solved by application of Gears method. This paper describes the transformer model and the necessary set of equations based on linearization of the iron cores saturable magnetic characteristic at each time interval. >

An application of nonlinear optimization in molecular biology

Abstract A maximum likelihood approach has been proposed for finding protein binding sites on strands of DNA [G.D. Stormo, G.W. Hartzell, Proceedings of the National Academy of Sciences of the USA 86 (1989) 1183]. We formulate an optimization model for the problem and present calculations with experimental sequence data to study the behavior of this site identification method.

A Mathematical Model of Breast Cancer Treatment with CMF and Doxorubicin

A mathematical model is presented to investigate the ordering phenomenon observed in the comparison of alternating to sequential regimens of CMF (cyclophosphamid, methotrexate, 5-fluorouracil) and doxorubicin used in breast cancer chemo-therapy. The ordinary differential equation model incorporates cell cycle specificity and resistance to study why doses of the same drugs given in different orders result in different clinical outcomes. The model employs a pulsing condition to simulate treatment and induced resistance, and we investigate treatment outcome by simulating a patient population by varying parameters using uniform distributions. The results of these simulations correspond to those observed in prior clinical studies and suggest that drug resistance might be a key mechanism in the sequential regimen’s superiority.

An ellipsoid algorithm for equality-constrained nonlinear programs

The ellipsoid algorithm is a simple method that yields accurate solutions to convex and many nonconvex nonlinear programming problems. Unfortunately, convergence of the method requires that the feasible set be of full dimension, so it cannot be used with equality constraints. This thesis presents a variant of the ellipsoid algorithm that solves convex problems having linear equality constraints with or without inequality constraints. The experimental results presented show that the new method is also effective for some problems that are nonconvex or that have nonlinear equality constraints.

An automated data collection system for membrane transport experiments : Part I. Test of onsager reciprocity

Abstract This work presents a direct experimental examination of the linear phenomenological flux equations from the thermodynamics of irreversible processes for membrane transport. Results are given which confirm Onsager reciprocity in a NaClH2O—anion exchange membrane system with concentration and hydrostatic pressure differences. Data for isothermal, non-steady-state experiments are collected from a computerized and automated membrane transport apparatus. For the first time, the solvent and solute flux equations are solved simultaneously, and the four phenomenological transport coefficients are obtained from a single experiment using an ellipsoid algorithm for non-linear programming. It appears that the dependence of the L coefficients on the logarithmic mean transmembrane concentration, cse, is more important than the role of the water flux as an indicator of the limit of the linear region for membrane transport processes. Only in those experiments where cse was held nearly constant was Onsager reciprocity obtained.

Computing the internal transient voltage response of a transformer with a nonlinear core using Gear's method. II. Verification

An EHV transformers insulation structure must be designed to withstand the internal electrical stresses generated during system transients. Computer models are employed for predetermination of these stresses. Part I of this paper developed a detailed transformer model and solution method representing the nonlinear, saturable characteristic of tire core during transients. The resulting set of stiff nonlinear differential algebraic equations are solved by application of Gears method. Part II presents the verification of this methodology. This is accomplished by comparing the computed and measured response of a 765/345/34.5 kV 500 MVA autotransformer during energization and transient excitation. >

A Mathematical Model of Cell Cycle Effects in Gastric Cancer Chemotherapy

A mathematical model is presented to investigate the relationship between drug order and treatment response in gastric cancer chemotherapy involving a taxane (either paclitaxel or docetaxel) coupled with flavopiridol. To model treatment effects, we simulate treatment by bolus injection and employ a pulsing condition to indicate cell kill as well as instantaneous changes to the cell’s transition rates. Cell population growth is described using an ordinary differential equation model whereby we examine the treatment effects upon cells in various stages of the cell cycle. Ultimately, the results generated support prior clinical investigations which indicate that for an enhanced synergistic effect, flavopiridol must be administered following taxane therapy.

The peak sidelobe distribution for binary codes

Low aperiodic-autocorrelation peak sidelobe levels (PSLs) relate to enhanced range resolution for binary-phase-coded radar and communication waveforms. Typical methods to identify the minimum-attainable PSL for a given code length N require exhaustive calculations whose computational burden grows exponentially with N. In this project, exact PSL histograms were determined for computationally practical lengths. These histograms may lead to ways to estimate PSL distributions for computationally impractical lengths. Plots of the lower four moments for N between 1 and 45 showed that the moments can be approximated closely by aNk. Histograms for N = 45 were compared to a binomially-distributed PSL PDF model based on statistically independent sidelobes. The independent-sidelobe model agreed closely with truth for middle-to high-PSL values, and only varied significantly from truth for PSLs one or two units away from the lowest achievable PSL. Future work will examine ways to develop the PDF from the moments accurately enough to estimate minimum PSL for a given N, and ways to account for sidelobe dependence in the probabilistic model.