In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.

A function f in one variable de ned on an interval I R is convex if f 00(x) 0 for all x 2I, and concave if f 00(x) 0 for all x 2I. The graph of convex and concave function have the following shapes: Convex: [Concave: \ If f is a quadratic form in one variable, it can be written as f (x) = ax2. In this case, f is convex if a 0 and concave if a 0.

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In the previous couple of lectures, we’ve been focusing on the theory of convex sets. In this lecture, we shift our focus to the other important player in convex optimization, namely, convex functions. Here are some of the topics that we will touch upon: Convex, concave, strictly convex, and strongly convex functions

2022年12月7日 · NAME OOQP - A package for solving convex quadratic programming problems. SYNOPSIS This directory contains OOQP, a package for solving convex quadratic programming problems (QP). These are optimization problems in which the objective function is a convex qu

Theorem 1.10 (Jensen’s Inequality). For a convex function fon the interval I, let x 1;x 2; ;x n2Iand 1; 2; ; n2(0;1) satisfying P n j=1 j = 1. Then f( 1x 1 + + nx n) 1f(x 1) + + nf(x n): When f is strictly convex, equality sign in this inequality holds if and only if x 1 = x 2 = = x n. Perhaps we need to explain why the linear combination is ...

This is just a quick and condensed note on the basic definitions and characterizations of concave, convex, quasiconcave and (to some extent) quasiconvex functions, with some examples. Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have:

Convex functions have another obvious property, which is related to the location of the tangent to the graph of the function. The function \(f\left( x \right)\) is convex downward on the interval \(\left[ {a,b} \right]\) if and only if its graph does not lie below the tangent drawn to it at any point \({x_0}\) of the segment \(\left[ {a,b ...

In applications, we encounter many constrained optimization problems. Examples. The constrained can be a convex set C. That is. 0 if x 2 C +1 otherwise . For more applications, see Boyd’s book. Here, fi’s are convex. The space X is a Hilbert space. Here, we just take X = RN.

Almost every convex function can be expressed as the pointwise supremum of a family of affine functions. / h = ∑ on is concave for 0 1, and its extension is nondecreasing. If is concave and...