Linear programming Lecturer : Michel Goemans 1 Basics
نویسنده
چکیده
Linear Programming deals with the problem of optimizing a linear objective function subject to linear equality and inequality constraints on the decision variables. Linear programming has many practical applications (in transportation, production planning, ...). It is also the building block for combinatorial optimization. One aspect of linear programming which is often forgotten is the fact that it is also a useful proof technique. In this first chapter, we describe some linear programming formulations for some classical problems. We also show that linear programs can be expressed in a variety of equivalent ways.
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Lectures 9 and 10 Lecturer : Michel
1 Semidefinite programming Let Sn×n be the set of n by n real symmetric matrices. Definition 1 A ∈ Sn×n is called positive semidefinite, denoted A 0, if xAx ≥ 0 for any x ∈ R. There are several well-known equivalent ways to state positive semidefiniteness. Proposition 1 The following are equivalent: (i) A is positive semidefinite. (ii) Every eigenvalue of A is nonnegative. (iii) There is a matr...
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