Lecture

Linear Optimization: Directional Derivatives

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In course

MATH-431: Theory of stochastic calculus

Introduction to the mathematical theory of stochastic calculus: construction of the Ito stochastic integral, proof of the Ito formula, introduction to stochastic differential equations, Girsanov's theorem and the Feynman-Kac formula, the martingale represe ...

Description

This lecture covers the concept of directional derivatives in linear optimization, focusing on the impact of considering different directions on the objective function. It explains how to optimize a function along a specific direction and the implications of non-linearly independent directions.

Instructors (2)

Robert Dalang

Robert Dalang, né en 1961, a reçu le diplôme de Mathématicien-EPFL en 1983 et est lauréat du Prix Dommer. Il passe l'année 1985-86 à Cornell University (USA) comme chercheur invité. Il obtient le doctorat au Département de mathématiques de l'EPFL en 1987. Son domaine de spécialisation est la théorie des processeurs stochastiques. En 1987, Robert Dalang est nommé professeur assistant au Département de statistiques de l'Université de Californie à Berkeley (USA). En 1988, il reçoit une bourse post-doctorale du Fonds national scientifique américain et effectue des recherches sur les propriétés markoviennes de processus stochastiques à plusieurs paramètres. En 1990, il est nommé à Tufts University (Boston, USA). Il est promu professeur associé en mai 1993. Une partie importante de ses recherches se font dans le cadre de contrats avec le Fonds national scientifique américain et l'Office de la recherche de l'armée américaine. En collaboration avec le Prof. R. Cairoli du Département de mathématiques de l'EPFL, il a écrit un livre sur l'optimisation stochastique séquentielle publié en 1996 aux éditions John Wiley. M. Dalang est nommé professeur extraordinaire de probabilités au Département de mathématiques en 1995. Il y poursuit des travaux de recherche en processus stochastiques et probabilité appliquée et participe à l'enseignement des processus stochastiques, de la théorie des probabilités et des cours de mathématiques aux sections d'ingénieurs. Il dirige régulièrement des thèses de doctorats, est éditeur de plusieurs journaux de recherche mathématique et travail en collaboration avec des chercheurs de plusieurs universités européennes et américaines.

Thomas Mountford

Teaching & PhD Past EPFL PhD Students Damien Gauthier, Samir Brahim Belhaouari, Driss Baraka, Daniel Rodrigues Valesin, Samuel Schöpfer, Kamran Kalbasi, Jacques Saliba, Hao Xue, Keming Chen Past EPFL PhD Students as codirector Guillaume Charles Woessner Courses Analysis I MATH-101(d) Study the fundamental concepts of analysis, and the differential and integral calculus of real-valued functions of one real variable. Markov chains MATH-332 The course follows the text of Norris

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Related concepts (28)

Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

Convex optimization

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

Combinatorial optimization

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

Multi-objective optimization

Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective is a type of vector optimization that has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.

Constrained optimization

In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized.

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