Reha H. Tütüncü

SDPT3 — A Matlab software package for semidefinite programming, Version 1.3

This invention relates to stabilizing compositions; to the process for preparing a novel halogen-containing polymer; and to polymers stabilized against the deteriorative effect of heat which comprises a vinyl chloride or vinylidene chloride homopolymer or copolymer and a stabilizing amount of: as a first stabilizer an organotin halide exhibiting the formula RSnX3 wherein R is a hydrocarbon and X is halogen and as a second stabilizer a sulfur-containing organotin compound exhibiting two direct carbon to tin bonds and two direct sulfur to tin bonds.

Solving semidefinite-quadratic-linear programs using SDPT3

Abstract. This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primal-dual path-following algorithms. The software developed by the authors uses Mehrotra-type predictor-corrector variants of interior-point methods and two types of search directions: the HKM and NT directions. A discussion of implementation details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.

Robust Asset Allocation

This article addresses the problem of finding an optimal allocation of funds among different asset classes in a robust manner when the estimates of the structure of returns are unreliable. Instead of point estimates used in classical mean-variance optimization, moments of returns are described using uncertainty sets that contain all, or most, of their possible realizations. The approach presented here takes a conservative viewpoint and identifies asset mixes that have the best worst-case behavior. Techniques for generating uncertainty sets from historical data are discussed and numerical results that illustrate the stability of robust optimal asset mixes are reported.

On the Nesterov--Todd Direction in Semidefinite Programming

We study different choices of search direction for primal-dual interior-point methods for semidefinite programming problems. One particular choice we consider comes from a specialization of a class of algorithms developed by Nesterov and Todd for certain convex programming problems. We discuss how the search directions for the Nesterov--Todd (NT) method can be computed efficiently and demonstrate how they can be viewed as Newton directions. This last observation also leads to convenient computation of accelerated steps, using the Mehrotra predictor-corrector approach, in the NT framework. We also provide an analytical and numerical comparison of several methods using different search directions, and suggest that the method using the NT direction is more robust than alternative methods.

A reduced space interior point strategy for optimization of differential algebraic systems

Abstract A novel nonlinear programming (NLP) strategy is developed and applied to the optimization of differential algebraic equation (DAE) systems. Such problems, also referred to as dynamic optimization problems, are common in process engineering and remain challenging applications of nonlinear programming. These applications often consist of large, complex nonlinear models that result from discretizations of DAEs. Variables in the NLP include state and control variables, with far fewer control variables than states. Moreover, all of these discretized variables have associated upper and lower bounds that can be potentially active. To deal with this large, highly constrained problem, an interior point NLP strategy is developed. Here a log barrier function is used to deal with the large number of bound constraints in order to transform the problem to an equality constrained NLP. A modified Newton method is then applied directly to this problem. In addition, this method uses an efficient decomposition of the discretized DAEs and the solution of the Newton step is performed in the reduced space of the independent variables. The resulting approach exploits many of the features of the DAE system and is performed element by element in a forward manner. Several large dynamic process optimization problems are considered to demonstrate the effectiveness of this approach, these include complex separation and reaction processes (including reactive distillation) with several hundred DAEs. NLP formulations with over 55 000 variables are considered. These problems are solved in 5–12 CPU min on small workstations.

An Interior-Point Method for a Class of Saddle-Point Problems

We present a polynomial-time interior-point algorithm for a class of nonlinear saddle-point problems that involve semidefiniteness constraints on matrix variables. These problems originate from robust optimization formulations of convex quadratic programming problems with uncertain input parameters. As an application of our approach, we discuss a robust formulation of the Markowitz portfolio selection model.

Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity

Abstract We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, is a convex quadratic or semidefinite programming problem, depending on the formulation. Both of these problems can be efficiently solved by numerical optimization software. In the quadratic programming formulation the positivity of the risk-neutral pdf is heuristically handled by posing linear inequality constraints at the spline nodes. In the other approach, this property of the risk-neutral pdf is rigorously ensured by using a semidefinite programming characterization of nonnegativity for polynomial functions. We tested our approach using data simulated from Black–Scholes option prices and using market data for options on the S&P 500 Index. The numerical results we present show the effectiveness of our methodology for estimating the risk-neutral probability density function.

Satisfying convex risk limits by trading

Abstract.A random variable, representing the final position of a trading strategy, is deemed acceptable if under each of a variety of probability measures its expectation dominates a floor associated with the measure. The set of random variables representing pre-final positions from which it is possible to trade to final acceptability is characterized. In particular, the set of initial capitals from which one can trade to final acceptability is shown to be a closed half-line

On the implementation of SDPT3 (version 3.1) - a MATLAB software package for semidefinite-quadratic-linear programming

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An Infeasible-Interior-Point Potential-Reduction Algorithm for Linear Programming

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