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Lecture

Linear Programming Duality

In course

MATH-261: Discrete optimization

This course is an introduction to linear and discrete optimization. Warning: This is a mathematics course! While much of the course will be algorithmic in nature, you will still need to be able to prove theorems. ...

Description

This lecture covers the concept of duality in linear programming, exploring the relationship between primal and dual problems. Starting with an introduction to duality, the instructor explains the duality theorem and its implications. The lecture then delves into the interpretation of dual variables and complementary slackness conditions. Various examples are provided to illustrate the application of duality in optimizing linear programming problems.

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Instructors (2)

Rom Pinchasi

Friedrich Eisenbrand

Friedrich Eisenbrand's main research interests lie in the field of discrete optimization, in particular in algorithms and complexity, integer programming, geometry of numbers, and applied optimization. He is best known for his work on efficient algorithms for integer programming in fixed dimension and the theory of cutting planes, which are an important tool to solve large scale industrial optimization problems in practice. Before joining EPFL in March 2008, Friedrich Eisenbrand was a full professor of mathematics at the University of Paderborn. Friedrich received the Heinz Maier-Leibnitz award of the German Research Foundation (DFG) in 2004 and the Otto Hahn medal of the Max Planck Society in 2001.

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Related concepts (32)

Linear programming

Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

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.

Color

Color (American English) or colour (Commonwealth English) is the visual perception based on the electromagnetic spectrum. Though color is not an inherent property of matter, color perception is related to an object's light absorption, reflection, emission spectra and interference. For most humans, color are perceived in the visible light spectrum with three types of cone cells (trichromacy). Other animals may have a different number of cone cell types or have eyes sensitive to different wavelength, such as bees that can distinguish ultraviolet, and thus have a different color sensitivity range.

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.

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