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Unformatted text preview: and x ∈ Rn is the variable. For simplicity, we will assume that
dom fi = Rn for each i.
Now introduce a new scalar variable t ∈ R and form the convex problem
minimize cT x
subject to tfi (x/t) ≤ 0, i = 1, . . . , m,
Ax = b, t = 1.
Deﬁne
K = cl{(x, t) ∈ Rn+1  tfi (x/t) ≤ 0, i = 1, . . . , m, t > 0}.
Then our original problem can be expressed as
minimize cT x
subject to (x, t) ∈ K,
Ax = b, t = 1.
This is a conic problem when K is proper.
You will relate some properties of the original problem to K .
(a) Show that K is a convex cone. (It is closed by deﬁnition, since we take the closure.)
(b) Suppose the original problem is strictly feasible, i.e., there exists a point x with fi (x) < 0,
¯
i = 1, . . . , m. (This is called Slater’s condition.) Show that K has nonempty interior.
(c) Suppose that the inequalities deﬁne a bounded set, i.e., {x  fi (x) ≤ 0, i = 1, . . . , m} is
bounded. Show that K is pointed.
3.16 Exploring nearly optimal points. An optimization algorithm will ﬁnd an optimal point for a problem,...
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 Fall '13
 F.Borrelli
 The Aeneid

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