BoydC3pt3

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Quasiconvex functions

Definition and examples

Def 4.1. \(f:\mathbb{R}^n\to\mathbb{R}\) is quasiconvex if \(\mathbf{dom} f\) is convex and the sublevel sets

\[S_{\alpha} = \{x\in\mathbf{dom}~f~|~f(x)\leq\alpha\} \]

are convex for all \(\alpha\).

\(f\) is quasiconcave if \(−f\) is quasiconvex.

\(f\) is quasilinear if it is quasiconvex and quasiconcave.

Basic propertie

Thm 4.1. modified Jensen inequality: for quasiconvex \(f\)

\[f(\theta x+(1-\theta)y)\leq\max\{f(x),f(y)\} \]

Differentiable quasiconvex functions

Thm 4.2. first-order condition: differentiable \(f\) with cvx domain is quasiconvex iff

\[f(y)\leq f(x)\Rightarrow \nabla f(x)^T(y-x)\leq 0 \]

Thm 4.3. Second-order conditions: If \(f\) is quasiconvex

\[y^T\nabla f(x) = 0 \Rightarrow y^T\nabla^2 f(x)y\geq 0 \]

Operations that preserve quasiconvexity

• nonnegative weighted maximum: \(f=\max\{\omega_1f_1,\cdots,\omega_m f_m\}\)
• pointwise supremum: \(f(x)=\sup_{y\in C}(\omega(y)f(x,y))\)
• composition: \(g\) quasiconvex and \(h\) is nondecreasing \(\Rightarrow\) \(f=h\circ g\) is quasiconvex.
• If \(f\) is quasiconvex, \(g(x)=f(Ax+b)\) and \(\tilde{g}(x)=f((Ax+b)/(c^Tx+b))\) are quasiconvex.
• minimization: \(f(x, y)\) is quasiconvex jointly in \(x\) and \(y\), \(C\) is convex set. \(g(x)=\inf_{y\in C}f(x,y)\) is quasiconvex.

Rmk. sums of quasiconvex functions are not necessarily quasiconvex

Log-concave and log-convex functions

Definition

Def 5.1. A positive function \(f\) is log-convex if \(\log f\) is convex:

\[f(\theta x+(1-\theta)y)\leq f(x)^{\theta}f(y)^{\theta},\quad 0\leq\theta\leq 1 \]

\(f\) is log-concave if \(\log f\) is concave

Rmk. log-convex \(\Rightarrow\) convex, but concave \(\Rightarrow\) log-concave

Properties

Thm 5.1. \(f\) with convex domain is log-concave iff

\[f(x)\nabla^2f(x)\preceq\nabla f(x)\nabla f(x)^T \]

Thm 5.2. operations preserve log-convex/log-concave

• product of log-concave(convex) functions is log-concave(convex)
• sum of log-convex functions is log-convex, but sum of log-concave functions is not always log-concave.
• If \(f(x,y)\) is log-convex for \(x\), then

\[g(x) = \int_C f(x,y) dy \]

is log convex.

• integration: if \(f:\mathbb{R}^n\times \mathbb{R}^m\to\mathbb{R}\) is log-concave, then

\[g(x) = \int f(x,y) dy \]

is log-convave.

Ex. If \(f\) and \(g\) are log-concave on \(\mathbb{R}^n\), then so is the convolution

\[(f*g)(x) = \int f(x-y)g(y)dy \]

If \(C\subseteq\mathbb{R}^n\) convex and \(y\) is a random variable with log-concave pdf then

\[f(x) = \mathbf{prob}~(x+y\in C) \]

is log-concave.

Convexity w.r.t. generalized inequalities

Def 6.1. \(f:\mathbb{R}^n\to\mathbb{R}^m\) is \(K\)-convex if \(\mathbf{dom}~f\) is convex and

\[f(\theta x+(1-\theta)y) \preceq_K \theta f(x) + (1-\theta)f(y) \]

for \(x,y\in \mathbf{dom}~f\), \(0\leq\theta\leq 1\).