Maria Cristina Recchioni

Box-constrained multi-objective optimization: A gradient-like method without “a priori” scalarization

Abstract The aim of this paper is the development of an algorithm to find the critical points of a box-constrained multi-objective optimization problem. The proposed algorithm is an interior point method based on suitable directions that play the role of gradient-like directions for the vector objective function. The method does not rely on an “a priori” scalarization and is based on a dynamic system defined by a vector field of descent directions in the considered box. The key tool to define the mentioned vector field is the notion of vector pseudogradient. We prove that the limit points of the solutions of the system satisfy the Karush–Kuhn–Tucker (KKT) first order necessary condition for the box-constrained multi-objective optimization problem. These results allow us to develop an algorithm to solve box-constrained multi-objective optimization problems. Finally, we consider some test problems where we apply the proposed computational method. The numerical experience shows that the algorithm generates an approximation of the local optimal Pareto front representative of all parts of optimal front.

The use of the Pontryagin maximum principle in a furtivity problem in time-dependent acoustic obstacle scattering

Abstract In this paper we consider a furtivity problem in the context of time-dependent three-dimensional acoustic obstacle scattering. The scattering problem for a ‘passive’ obstacle is the following: an incoming acoustic wavepacket is scattered by a bounded simply connected obstacle with locally Lipschitz boundary having a known boundary acoustic impedance. The scattered wave is the solution of an exterior problem for the wave equation. To make the obstacle furtive we leave ‘passive’ obstacles and we consider ‘active’ obstacles, that is obstacles that, when hit by the incoming wavepacket, react with a pressure current circulating on their boundary. The furtivity problem consists of making the acoustic field scattered by the obstacle ‘as small as possible’ by choosing a control function, that is a pressure current on the boundary of the obstacle, in the function space of the admissible controls. It consists of finding the control function that minimizes a cost functional that will be made precise later. This furtivity problem is of great relevance in many applications. The mathematical model for this furtivity problem is a control problem for the wave equation. In the boundary condition for the wave equation on the boundary of the obstacle we introduce a control function, the so-called pressure current. The cost functional depends on the control function, and on the scattered acoustic field. Note that the scattered field depends on the control function via the boundary conditions. Using the Pontryagin maximum principle we show that, for a suitable choice of the cost functional, the first-order optimality conditions for the furtivity problem considered can be formulated as an exterior problem defined outside the obstacle for a system of two coupled wave equations. This is the main purpose of the paper. Moreover, to solve this exterior problem numerically we develop a highly parallelizable method based on a ‘perturbative series’ of the type proposed in 1. This method obtains the time-dependent scattered field and the control function as superpositions of time harmonic functions. The space-dependent parts of each time harmonic component of the scattered field and of the control function are obtained by solving an exterior boundary value problem for two coupled Helmholtz equations. The mathematical model and the numerical method proposed are validated by studying some test problems numerically. The results obtained with a parallel implementation of the numerical method proposed on the test problems are shown and discussed from the numerical and the physical point of view. The quantitative character of the results obtained is established. Animations (audio, video) relative to the numerical experiments can be found at stacks.iop.org/WRM/11/549. MThis article features online multimedia enhancements

A Quadratically Convergent Method for Linear Programming

Abstract A new method to solve linear programming problems is introduced. This method follows a path defined by a system of o.d.e., and for nondegenerate problem is quadratically convergent.

A masking problem in time dependent acoustic obstacle scattering

A masking problem in time dependent three dimensional acoustic obstacle scattering is considered. The masking problem consists in making masked a bounded scatterer characterized by an acoustic boundary impedance and immersed in a homogeneous isotropic medium that, when hit by an incident acoustic field, generates a scattered acoustic field. The precise definition of the masking problem is given later. This problem has been formulated as an optimal control problem for the wave equation. The corresponding first order optimality condition is derived and solved with a highly parallelizable numerical method. Some numerical experience on a test problem is shown.

The time harmonic electromagnetic field in a disturbed half‐space: An existence theorem and a computational method

The scattering of time harmonic electromagnetic waves by a perfectly conducting surface that is the boundary of a ‘‘disturbed half‐space’’ is considered. This problem is translated in a boundary value problem for an elliptic system of partial differential equations. Under appropriate hypotheses an existence theorem and an integral representation formula for the solution of this boundary value problem is given. Based on this integral representation formula a new method to compute the solution of the boundary value problem is proposed. This method involves only quadratures and is fully parallelizable. Finally some numerical examples of the results obtained on test problems with this computational method are shown.

Time harmonic electromagnetic scattering from a bounded obstacle: An existence theorem and a computational method

Let Ω⊂R3 be a bounded simply connected obstacle with boundary ∂Ω locally Lipschitz, we consider the scattering of a time harmonic electromagnetic wave that hits Ω when ∂Ω is assumed to be perfectly conducting. The scattered electromagnetic field is the solution of an exterior boundary value problem for the vector Helmholtz equation. Under suitable hypotheses we prove the existence and uniqueness of the solution of this boundary value problem and we give a new numerical method to compute this solution. The numerical method proposed is based on a perturbative series and is highly parallelizable. Some numerical results obtained with the numerical method proposed on test problems are presented and discussed from the numerical and the physical point of view.

The Analysis of Real Data Using a Multiscale Stochastic Volatility Model

In this paper we use filtering and maximum likelihood methods to solve a calibration problem for a multiscale stochastic volatility model. The multiscale stochastic volatility model considered has been introduced in Fatone et al. (2009), generalises the Heston model and describes the dynamics of the asset price using as auxiliary variables two stochastic variances on two different time scales. The aim of this paper is to estimate the parameters of this multiscale model (including the risk premium parameters when necessary) and its two initial stochastic variances from the knowledge, at discrete times, of the asset price and, eventually, of the prices of call and/or put European options on the asset. This problem is translated into a maximum likelihood problem with the likelihood function defined through the solution of a filtering problem. Furthermore we develop a tracking procedure that is able to track the asset price and the values of its two stochastic variances for time values where there are no data available. Numerical examples of the solution of the calibration problem and of the performance of the tracking procedure using high frequency synthetic data and daily real data are presented. The real data studied are two time series of electric power price data taken from the US electricity market and the 2005 data relative to the US S&P 500 index and to the prices of a call and a put European option on the US S&P 500 index. The calibration procedure is applied to these data and the results of the calibration are used in the tracking procedure to forecast the asset and option prices. The forecasts of the asset prices and of the option prices are compared with the prices actually observed. This comparison shows that the forecasts are of very high quality even when we consider ‘spiky’ electric power price data.

An explicitly solvable Heston model with stochastic interest rate

This paper deals with a variation of the Heston hybrid model with stochastic interest rate illustrated in Grzelak and Oosterlee (2011). This variation leads to a multi-factor Heston model where one factor is the stochastic interest rate. Specifically, the dynamics of the asset price is described through two stochastic factors: one related to the stochastic volatility and the other to the stochastic interest rate. The proposed model has the advantage of being analytically tractable while preserving the good features of the Heston hybrid model in Grzelak and Oosterlee (2011) and of the multi-factor Heston model in Christoffersen et al. (2009). The analytical treatment is based on an appropriate parametrization of the probability density function which allows us to compute explicitly relevant integrals which define option pricing and moment formulas. The moments and mixed moments of the asset price and log-price variables are given by elementary formulas which do not involve integrals. A procedure to estimate the model parameters is proposed and validated using three different data-sets: the prices of call and put options on the U.S. S&P 500 index, the values of the Credit Agricole index linked policy, Azione Piu Capitale Garantito Em.64., and the U.S. three-month, two and ten year government bond yields. The empirical analysis shows that the stochastic interest rate plays a crucial role as a volatility factor and provides a multi-factor model that outperforms the Heston model in pricing options. This model can also provide insights into the relationship between short and long term bond yields.

An interior point algorithm for global optimal solutions and KKT points

In this paper some theorems which characterize the global optimal solutions of nonlinear programming problems are proved. Two algorithms are derived using these results. The first is a path following algorithm to approximate the Karush Kuhn Tucker points of linearly constrained optimization problems and the second is an algorithm to solve linearly constrained global optimization problems. The convergence of these algorithms is proved under suitable assumptions. Numerical results obtained on several test problems are shown.

Some Control Problems for the Maxwell Equations Related to Furtivity and Masking Problems in Electromagnetic Obstacle Scattering

Furtivity and masking problems in time dependent electromagnetic obstacle scattering are formulated as control problems for the Maxwell equations. An ingenious way of using the Pontryagin maximum principle makes possible to express the corresponding first order optimality conditions as a system of partial differential equations. This fact offers relevant computational advantages, such that the use of iterative optimization methods in the solution of the optimal control problems can be avoided and highly parallelizable numerical methods can be developed to solve the control problems proposed.