On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes

On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes

Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations

Imitation Games and Computation

TAn imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths.

A second derivative SQP method: local convergence

In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact ℓ1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud \ud Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk—a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1-penalty function over a sequence of increasing values of the penalty parameter.\ud \ud Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set

Sequential Convex Programming Methods for Solving Nonlinear Optimization Problems with DC constraints

This paper investigates the relation between sequential convex programming (SCP) as, e.g., defined in [24] and DC (difference of two convex functions) programming. We first present an SCP algorithm for solving nonlinear optimization problems with DC constraints and prove its convergence. Then we combine the proposed algorithm with a relaxation technique to handle inconsistent linearizations. Numerical tests are performed to investigate the behaviour of the class of algorithms.Comment: 18 pages, 1 figur