Lec04 Convex optimization problems¶
1.Optimization problem 优化问题¶
1.1Optimization problem in standard form¶
option value¶
\[p^*= inf\{f_0(x) | f_i(x) ≤ 0, i = 1, . . . , m, h_i(x) = 0, i = 1, . . . , p\}\]
如果问题不可行，取 p* 为 ∞
如果问题无下界，取 p* 为-∞
如果x可行且 \(f_i(x)=0\)，我们称约束 \(f_i(x)\leq 0\) 的第i个不等式x处起作用
1.2 Optimal and locally optimal points¶
feasible¶
x is feasible if \(x ∈ dom f_0\) and it satisfies the constraints
a feasible x is optimal if \(f_0(x) = p^*\);\(X_{opt}\) is the set of optimal points
local optimal¶
x is local optimal if there is an R>0 such that x is optimal for
1.3 Implict constraints¶
we call the domain of the problem is
2.Convex optimization 凸优化¶
2.1 standard form convex optimization problem¶
important property:
- feasible set of a convex optimization problem is convex
2.2 Local and global optima¶
In fact
- any locally optimal point of a convex problem is (globally) optimal
2.3 Optimality criterion for differentiable f0¶
x is optimal if and only if it is feasible and
\[
∇f_0(x)^T (y - x) ≥ 0
\]
2.4 Equivalent convex problems¶
Equivalent convex problems¶
introducing equality constraints¶
introducing slack variables for linear inequalities¶
epigraph form¶
minimizing over some variables(partial optimate)¶
2.5 Quasiconvex optimization¶
when the \(f_0\) is quaisconvex ,can have locally optimal points that are not globally optimal
solution¶
3. Linear program 线性规划问题¶
\[
minimize \ c^Tx+d
\]
\[
subject \ to \ Gx≼ h
\]
\[
Ax=b
\]
feasible set is a polyhedron.
Linear-fractional program 线性分式规划问题¶
4.Quadratic program二次优化问题¶
QP¶
目标函数为凸二次型，且约束函数为仿射。
QCQP¶
不等式约束也是凸二次型，则是QCQP
Sencond-order cone programming (SOCP)¶
5.Geometric programming几何规划¶
monimal function and posyminal function¶
GP¶
GP一般来说都不是凸问题，在此情况下可以转化为凸优化问题
solution¶
\[
y_i=log\ x_i
\]
6.Generalized inequality constraints广义不等式约束¶
convex problem with generalized inequality constraints¶
conic form problem¶
Semidefinite program (SDP)¶
7.Vector optimization向量优化¶
general and convex vector optimization problem¶
