Michel Denault

Protein Library Design and Screening

In designing protein libraries for selection, we must coordinate our capacity to create a large diversity of protein variants with the physical limitations of what we can actually screen. This chapter aims to bring the language of probabilities into the protein engineers laboratory to answer some of our common questions: How can we most efficiently design a library? What fraction of the theoretical library diversity have we actually sampled at the end of the day? What is the probability of missing an individual of the library? Are the mutations present in the variants we have selected statistically meaningful or the product of random variation? The computation of these criteria throughout the process of experimental protein engineering will enable us to better design and evaluate the products of our libraries of protein variants.

A simulation-and-regression approach for stochastic dynamic programs with endogenous state variables

We investigate the optimum control of a stochastic system, in the presence of both exogenous (control-independent) stochastic state variables and endogenous (control-dependent) state variables. Our solution approach relies on simulations and regressions with respect to the state variables, but also grafts the endogenous state variable into the simulation paths. That is, unlike most other simulation approaches found in the literature, no discretization of the endogenous variable is required. The approach is meant to handle several stochastic variables, offers a high level of flexibility in their modeling, and should be at its best in non time-homogenous cases, when the optimal policy structure changes with time. We provide numerical results for a dam-based hydropower application, where the exogenous variable is the stochastic spot price of power, and the endogenous variable is the water level in the reservoir.

The Analytic-Center Cutting-Plane Method for Variational Inequalities: A Quadratic-Cut Approach

We introduce a cutting-plane, analytic-center algorithm for strongly monotone variational inequalities (VIs). The approach extends that of Goffin et al. (1997) and Denault and Goffin (1999). The VI is still treated as a convex feasibility problem, with linear cuts progressively shrinking alocalization set that contains the solution of the VI. However, a quadratic cut is used to improve the positioning of the point at which the next cut will be generated. Our approach uses quadratic, ellipsoidal cuts, based on the symmetrized Jacobian of the VI. Since it cannot be guaranteed that such quadratic cuts do not cut off the solution of the VI, they are used only for direction, and are not integrated as such in the localization set; only the linear part of the quadratic cuts can safely be added to the localization set. The introduction of the quadratic cut together with the drop of the quadratic part of the previous cut is studied carefully. Numerical results are given that illustrate the substantial improvement that quadratic cuts can yield over linear cuts.

Solving variational inequalities with a quadratic cut method: a primal-dual, Jacobian-free approach

We extend in two directions the Analytic Center, Cutting Plane Method for Variational Inequalities with quadratic cuts, ACCPM-VI(quadratic cuts), introduced by Denault and Goffin in 1998. First, we define a primal-dual method to find the analytic center at each iteration. Second, the Broyden-Fletcher-Goldfarb-Shanno Jacobian approximation, of quasi-Newton fame, is used in the definition of the cuts, making the algorithm applicable to problems without tractable Jacobians. The algorithm is tested on a variety of variational inequality problems, including one challenging problem of pricing the pollution permits put forward in the Kyoto Protocol.

Dynamic portfolio choices by simulation-and-regression

Simulation-and-regression methods have been recently proposed to solve multi-period, dynamic portfolio choice problems. In the constant relative risk aversion (CRRA) framework, the value function recursion vs portfolio weight recursion issue was previously examined in van Binsbergen and Brandt [24] and Garlappi and Skoulakis [14]. We revisit this issue in the context of an alternative simulation-and-regression algorithmic approach which does not rely on Taylor series approximations of the value function. We find that, in this context and for the CRRA example examined here, both approach are capable of obtaining precise results, but that the portfolio weight recursion variant of the algorithm provides more accurate results for a similar level of computational complexity, especially for problems with long maturities and large risk-aversion levels. HighlightsDynamic portfolio choices are computed with a simulation-and-regression approach.Unlike the current literature, our approach does not use Taylor series.We examine and compare two computational alternatives within this framework.We find that both alternatives can achieve precise results, one being more robust.

RESOLUTION NUMERIQUE DE PROBLEMES DE COMPLEMENTARITE LINEAIRE ET EVALUATION D'OPTIONS AMERICAINES

Abstract In the Black-Scholes framework, the American option pricing problem can be discretized into a finite-dimensional linear complementarity problem. We compare the numerical performances of three algorithms for the linear complementarity problem: the pivotal algorithms of Lemke and of Boriçi and Lüthi, and the iterative approach “Projected Successive Over-Relaxation”. We conclude that a special-purpose algorithm can provide performances vastly superior to those of generic algorithms.