Lecture

Simplex algorithm: From vertex to vertex

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Description

This lecture covers the simplex algorithm, starting with the problem in standard form and the basic ingredients needed for optimization. It explains the process of finding the next vertex by selecting descent directions and calculating distances to constraints. The lecture emphasizes the importance of selecting variables and provides details on identifying optimal vertices and dealing with unbounded or degenerate solutions.

In MOOC

Optimization: principles and algorithms - Linear optimization

What you'll learn Formulation: you will learn from simple examples how to formulate, transform and characterize an optimization problem. Constraints: you will learn how to represent the constraints of a linear optimization problem, both from a geometric and algebraic point of views. Duality: you will learn how to derive a companion problem called the "dual". Optimality conditions: you will learn sufficient and necessary conditions for an optimal solution. Simplex method: you will learn an algorithm to solve

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Related concepts (17)

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.

Simplex algorithm

In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial cones, and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope.

Program optimization

In computer science, program optimization, code optimization, or software optimization, is the process of modifying a software system to make some aspect of it work more efficiently or use fewer resources. In general, a computer program may be optimized so that it executes more rapidly, or to make it capable of operating with less memory storage or other resources, or draw less power. Although the word "optimization" shares the same root as "optimal", it is rare for the process of optimization to produce a truly optimal system.

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.

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.

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