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Lecture# Optimisation with Constraints: Interior Point Algorithm

Description

This lecture covers the optimization with constraints using the Karush-Kuhn-Tucker conditions and the interior point algorithm, focusing on two examples of quadratic programming. The KKT conditions and the Lagrangian are explored to find the optimal solutions.

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Related concepts (59)

MATH-212: Analyse numérique et optimisation

L'étudiant apprendra à résoudre numériquement divers problèmes mathématiques. Les propriétés théoriques de ces
méthodes seront discutées.

Paint is a liquid pigment that, after application to a solid material, and allowed to dry, adds a film-like layer to protect, add color, or provide texture. Paint can be made in many colors—and in many different types. Most paints are either oil-based or water-based, and each has distinct characteristics. For one, it is illegal in most municipalities to discard oil-based paint down household drains or sewers. Clean-up solvents are also different for water-based paint than oil-based paint.

Oil paint is a type of slow-drying paint that consists of particles of pigment suspended in a drying oil, commonly linseed oil. The viscosity of the paint may be modified by the addition of a solvent such as turpentine or white spirit, and varnish may be added to increase the glossiness of the dried oil paint film. The addition of oil or alkyd medium can also be used to modify the viscosity and drying time of oil paint. Oil paints were first used in Asia as early as the 7th century AD and can be seen in examples of Buddhist paintings in Afghanistan.

In algebra, a quadratic equation () is any equation that can be rearranged in standard form as where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.

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.

Constraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables, which is solved by constraint satisfaction methods. CSPs are the subject of research in both artificial intelligence and operations research, since the regularity in their formulation provides a common basis to analyze and solve problems of many seemingly unrelated families.

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