数学知识：Convex Optimization B

Convex optimization problems

• optimization in standard form
• convex optimization problems
• quasiconvex optimization
• linaer optimization
• quadratic optimization
• geometric programming
• generalized inequality constraints
• semidefinite programming
• vector optimization

optimization problem in standard form

\[\begin{align*} minimize~&f_0(x) \\ subject~to~&f_i(x)\le0,i=1,...,m \\ &h_i(x)=0, i=1,...,p \end{align*} \]

• \(x\in\bf{R}^n\) is the optimization variable

optimal and locally optimal points

\(x\) is feasible if \(x\in~dom~f_0\) and it satisfies the constraints

a feasible \(x\) is optimal if \(f_0(x)=p^*\); \(X_{opt}\) is the set of optimal points

x is locally optimal if there is an \(R>0\) such that \(x\) is optimal for

\[\begin{align*} minimize(over~z)~&f_0(z) \\ subject~to~&f_i(x)\le0,i=1,...,m~~~~h_i(z)=0,i=1,...,p \\ &\Vert z-x\Vert_2\le R \end{align*} \]

examples

implicit constraints

the standard form optimization problem has an implicit constraint

\[x\in\mathcal{D}=\bigcap\limits_{i=0}^m dom~f_i~\cap~\bigcap\limits_{i=1}^p dom~h_i, \]

• we call \(\mathcal{D}\) the domain of the problem

feasibility problem

\[\begin{align*} find~~~~~~~~~~&x \\ subject~to~&f_i(x)\le0,i=1,...,m \\ &h_i(x)=0,i=1,...,p \end{align*} \]

convex optimization problem

standard convex optimization problem

\[\begin{align*} minimize~&f_0(x) \\ subject~to~&f_i(x),i=1,...,m \\ &a_i^Tx=b_i,i=1,...,p \end{align*} \]

• \(f_0,f_1,..,f_m\) are convex; equality constraints are affine
• problem is quasiconvex if \(f_0\) is quasiconvex (and \(f_1,...,f_m\) convex)
  often written as

\[\begin{align*} minimize~&f_0(x) \\ subject~to~&f_i(x),i=1,...,m \\ &Ax=b \end{align*} \]

local and global optima

any locally optimal opint of a convex problem is (globally) optimal

optimality criterion for differentiable \(f_0\)

• unconstrainted problem

• equality constrainted problem

• minimization over nonnegative orthant

equivalent convex problem

• eliminating equality constraints

• introducing equality constraints

• introducing slack variables for linear inequalities

• epigraph form

• minimizing over some variables