Lecture

ALM with Inequalities: Next Steps in Optimization

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In course

MATH-329: Continuous optimization

This course introduces students to continuous, nonlinear optimization. We study the theory of optimization with continuous variables (with full proofs), and we analyze and implement important algorithms to solve constrained and unconstrained problems. ...

Description

This lecture covers the application of Augmented Lagrangian Method (ALM) with equality and inequality constraints in optimization problems. The instructor discusses the process of formulating and solving optimization problems with both types of constraints, emphasizing the importance of slack variables for handling inequalities. The lecture also delves into the concept of reformulating problems with slack variables to simplify the optimization process. Throughout the slides, various mathematical expressions and equations are used to illustrate the theoretical concepts behind ALM. The presentation concludes with a discussion on the practical implications of incorporating inequality constraints in optimization models.

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Instructor

Nicolas Boumal

Teaching & PhD PhD Students Andreea-Alexandra Musat Past EPFL PhD Students Christopher Criscitiello, Quentin Rebjock Courses Continuous optimization MATH-329 This course introduces students to continuous, nonlinear optimization. We study the theory of optimization with continuous variables (with full proofs), and we analyze and implement important algorithms to solve constrained and unconstrained problems. Linear Algebra MATH-111(e) The purpose of the course is to introduce

Official source

Ontological neighbourhood

Theoretical computer science

Algorithms and data structures: Combinatorial optimization

Related concepts (31)

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.

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.

Linear complementarity problem

In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968. Given a real matrix M and vector q, the linear complementarity problem LCP(q, M) seeks vectors z and w which satisfy the following constraints: (that is, each component of these two vectors is non-negative) or equivalently This is the complementarity condition, since it implies that, for all , at most one of and can be positive.

Mathematical economics

Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.

Constraint satisfaction

In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution through a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a set of values for the variables that satisfies all constraints—that is, a point in the feasible region. The techniques used in constraint satisfaction depend on the kind of constraints being considered.

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