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In 1994, Boyd and Laurent El Ghaoui, Eric Feron, and Ragu Balakrishnan authored the book Linear Matrix Inequalities in System & Control Theory. [15] Around 1999, he and Lieven Vandenberghe developed a PhD-level course and wrote the book Convex Optimization to introduce and apply convex optimization to other fields. [13]
In fact, while any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP, [8] it is known that there exist convex semialgebraic sets that are not representable by SDPs, that is, there exist convex semialgebraic sets that can not be written as a feasible region of a SDP. [9]
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).
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According to Boyd/Vandenberghe, which is considered a standard reference, a convex optimization problem has three additional requirements as compared to a general optimization problem, namely 1) the objective function must be convex (in the case of minimization), 2) the inequality constraint functions must be convex, and 3) the equality ...
In convex optimization, a linear matrix inequality (LMI) is an expression of the form ():= + + + + where = [, =, …,] is a real vector,,,, …, are symmetric matrices, is a generalized inequality meaning is a positive semidefinite matrix belonging to the positive semidefinite cone + in the subspace of symmetric matrices .
A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ f (called the affine minorants of f). References [ edit ]
In convex analysis, a non-negative function f : R n → R + is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality (+ ()) () for all x,y ∈ dom f and 0 < θ < 1.