A homotopy method for nonlinear semidefinite programming

In this paper, for solving the nonlinear semidefinite programming problem, a homotopy is constructed by using the parameterized matrix inequality constraint. Existence of a smooth path determined by the homotopy equation, which starts from almost everywhere and converges to a Karush–Kuhn–Tucker point, is proven under mild conditions. A predictor-corrector algorithm is given for numerically tracing the smooth path. Numerical tests with nonlinear semidefinite programming formulations of several control design problems with the data contained in COMPl _e ib are done. Numerical results show that the proposed algorithm is feasible and applicable.     

A homotopy continuation method for solving normal equations

In this paper, we present a continuation method for solving normal equations generated byC ² functions and polyhedral convex sets. We embed the normal map into a homotopyH, and study the existence and characteristics of curves inH ¹(0) starting at a specificd point. We prove the convergence of such curves to a solution of the normal equation under some conditions on the polyhedral convex setC and the functionf. We prove that the curve will have finite are length if the normal map, associated with the derivative df(·) and the critical coneK, is coherently oriented at each zero of the normal mapf _c inside a certain ball of ⁿ . © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.This research was performed at the Department of Industrial Engineering, University of Wisconsin-Madison, Madison, WI, USA.     

Option pricing in incomplete markets

Expected utility maximization is a very useful approach for pricing options in an incomplete market. The results from this approach contain many important features observed by practitioners. However, under this approach, the option prices are determined by a set of coupled nonlinear partial differential equations in high dimensions. Thus, it represents numerous significant difficulties in both theoretical analysis and numerical computations. In this paper, we present accurate approximate solutions for this set of equations.     

A homotopy perturbation method for fractional-order advection-diffusion-reaction boundary-value problems

In this paper we describe the application of the homotopy perturbation method (HPM) to two-point boundary-value problems with fractional-order derivatives of Caputo-type. We show that HPM is equivalent to the semi-analytical Adomian decomposition method when applied to a class of nonlinear fractional advection-diffusion-reaction models. A general expression is derived for the coefficients in the HPM series solution. Numerical experiments are given to demonstrate several properties of HPM, such as its dependence on the fractional order and the parameters in the model. In the case of more than one solution, HPM has difficulties to find the second solution in the model. Another example is given for which HPM seems to converge to a non-existing solution.     

A constraint shifting homotopy method for general non-linear programming

In this article, a constraint shifting homotopy method (CSHM) is proposed for solving non-linear programming with both equality and inequality constraints. A new homotopy is constructed, and existence and global convergence of a homotopy path determined by it are proven. All problems that can be solved by the combined homotopy interior point method (CHIPM) can also be solved by the proposed method. In contrast to the combined homotopy infeasible interior point method (CHIIPM), it needs a weaker regularity condition. And the starting point in the proposed method is not necessarily a feasible point or an interior point, so it is more convenient to be implemented than CHIPM and CHIIPM. Numerical results show that the proposed algorithm is feasible and effective.     

A constraint shifting homotopy method for convex multi-objective programming

In this paper, a constraint shifting combined homotopy method for solving multi-objective programming problems with both equality and inequality constraints is presented. It does not need the starting point to be an interior point or a feasible point and hence is convenient to use. Under some assumptions, the existence and convergence of a smooth path to an efficient solution are proven. Simple numerical results are given.     

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.     

A homotopy interior point method for semi-infinite programming problems

This paper presents a homotopy interior point method for solving a semi-infinite programming (SIP) problem. For algorithmic purpose, based on bilevel strategy, first we illustrate appropriate necessary conditions for a solution in the framework of standard nonlinear programming (NLP), which can be solved by homotopy method. Under suitable assumptions, we can prove that the method determines a smooth interior path from a given interior point to a point w ^*, at which the necessary conditions are satisfied. Numerical tracing this path gives a globally convergent algorithm for the SIP. Lastly, several preliminary computational results illustrating the method are given.     

A constraint shifting homotopy method for convex multi-objective programming

In this paper, a constraint shifting combined homotopy method for solving multi-objective programming problems with both equality and inequality constraints is presented. It does not need the starting point to be an interior point or a feasible point and hence is convenient to use. Under some assumptions, the existence and convergence of a smooth path to an efficient solution are proven. Simple numerical results are given.     

A noninterior point homotopy method for semi-infinite programming problems

In this paper, we present a new homotopy method which is a non-interior point homotopy method for solving semi-infinite programming problems. Under suitable assumptions, we prove that the method determines a smooth path from a given point. The new homotopy method generalizes the existing combined homotopy interior point method for semi-infinite programming problems to unbounded set, moreover, it is more convenient in that it enlarges the choice scope of the initial point. Some numerical examples are given to show its efficiency.     

A utility maximization approach to hedging in incomplete markets

In this paper we introduce the notion of portfolio optimization by maximizing expected local utility. This concept is related to maximization of expected utility of consumption but, contrary to this common approach, the discounted financial gains are consumed immediately. In a general continuous-time market optimal portfolios are obtained by pointwise solution of equations involving the semimartingale characteristics of the underlying securities price process. The new concept is applied to hedging problems in frictionless, incomplete markets.     

Option pricing in incomplete discrete markets

Various methods of option pricing in discrete time models are discussed. The classical risk minimization method often results in negative prices and a natural modification is proposed. Another method of risk minimization using an inductive procedure as in the Cox-Ross-Rubinstein model is also proposed. The definition of the risk interpreted as the maximum of possible loss is discussed.     

Stability of utility-maximization in incomplete markets

The effectiveness of utility-maximization techniques for portfolio management relies on our ability to estimate correctly the parameters of the dynamics of the underlying financial assets. In the setting of complete or incomplete financial markets, we investigate whether small perturbations of the market coefficient processes lead to small changes in the agent’s optimal behavior, as derived from the solution of the related utility-maximization problems. Specifically, we identify the topologies on the parameter process space and the solution space under which utility-maximization is a continuous operation, and we provide a counterexample showing that our results are best possible, in a certain sense. A novel result about the structure of the solution of the utility-maximization problem, where prices are modeled by continuous semimartingales, is established as an offshoot of the proof of our central theorem.     

Two-parameter homotopy method for nonlinear equations

We introduce a second auxiliary parameter into the zero-order deformation equation and propose a generalization of the homotopy analysis method. This includes the derivation of a general solution in terms of the Bell polynomials for nonlinear equations. Numerical examples show that the proposed zero-order deformation equation improves the convergence region and rate of the series solution and allows greater freedom in the selection of auxiliary operators. This facilitates the development of a homotopy iteration scheme for nonlinear equations with discontinuous or zero derivatives that are not amenable to Newton-type iteration schemes. The homotopy iteration scheme represents a generalization of conventional iteration schemes and additional examples demonstrate its applicability for a wider range of nonlinear problems.     

A note on the homotopy analysis method

The present work is devoted to using an analytic approach, namely the homotopy analysis method, to obtain convergent series solutions of strongly nonlinear problems. On the basis of the homotopy derivative concept described in Liao (2009) [3], a theorem is proved here which generalizes some lemmas and theorems provided in Liao (2009) [3] and Molabahrami and Khani (2007) [4]. Significant applicability of the theorem obtained here in some practical situations is demonstrated.     

A one-step optimal homotopy analysis method for nonlinear differential equations

In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity.     

A short overview of some behavioural scenarios for derivative pricing in incomplete markets

We shortly describe three different but related scenarios for determination of asset prices in an incomplete market: one scenario uses a market game approach whereas the other two are based on risk sharing or regret minimizing considerations. Furthermore, we point out some new dynamical schemes modeling the convergence of the buyer's and of the seller's prices of a given asset to a unique price. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A homotopy perturbation method for solving a neutron transport equation

The purpose of this paper consists in the finding of the solution for a stationary transport equation using the techniques of homotopy perturbation method (HPM). The results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

A new imputation method for incomplete binary data

In data analysis problems where the data are represented by vectors of real numbers, it is often the case that some of the data-points will have “missing values”, meaning that one or more of the entries of the vector that describes the data-point is not observed. In this paper, we propose a new approach to the imputation of missing binary values. The technique we introduce employs a “similarity measure” introduced by Anthony and Hammer (2006) [1]. We compare experimentally the performance of our technique with ones based on the usual Hamming distance measure and multiple imputation.