Optimization Models by Giuseppe C. Calafiore & Laurent el Ghaoui

Author:Giuseppe C. Calafiore & Laurent el Ghaoui [Calafiore, Giuseppe C. & Ghaoui, Laurent el]
Language: eng
Format: epub
Tags: Business & Economics, Electronics, Linear & Nonlinear Programming, Mathematics, Mechanical, Non-Fiction, Operations Research, Statistics, Technology & Engineering, Telecommunications
ISBN: 9781107050877
Google: rEilBAAAQBAJ
Publisher: Cambridge University Press
Published: 2014-10-31T00:00:00+00:00

which is an LP in the variable z, in conic standard form.

Remark 9.2 Geometric interpretation of LP. The set of points that satisfy the constraints of an LP (i.e., the feasible set) is a polyhedron (or a polytope, when it is bounded):

X = {x ∈ Rn : Aeqx = beq, Ax ≤ b}.

Let xf ∈ X be a feasible point. With such point is associated the objective level c⊤xf (from now on, we assume without loss of generality that d = 0). A point xf ∈ X is an optimal point, hence a solution of our LP, if and only if there is no other point x ∈ X with lower objective, that is:

xf ∈ X is optimal for LP ⇔ c⊤x ≥ c⊤xf, ∀y ∈ X,

see also the discussion in Section 8.4. Vice versa, the objective can be improved if one can find x ∈ X such that c⊤(x − xf) < 0. Geometrically, this condition means that there exists a point x in the intersection of the feasible set X and the open half-space {x : c⊤(x − xf) < 0}, i.e., that we can move away from xf in a direction that forms a negative inner product with direction c (descent direction), while maintaining feasibility. At an optimal point x∗ there is no feasible descent direction, see Figure 9.7. The geometric interpretation suggests that the following situations may arise in an LP.

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