A toy 2D optimization problem: geometric view via the epigraph form

L. El Ghaoui

A toy 2D optimization problem: geometric view via the epigraph form

Consider the toy problem

$\min _x 0.9 x_1^2-0.4 x_1 x_2-0.6 x_2^2-6.4 x_1-0.8 x_2:-1 \leq x_1 \leq 2, \quad 0 \leq x_2 \leq 3 .$

We can represent the problem in epigraph form, as

$\min _{x,t} t \geq 0.9 x_1^2-0.4 x_1 x_2-0.6 x_2^2-6.4 x_1-0.8 x_2:-1 \leq x_1 \leq 2, \quad 0 \leq x_2 \leq 3 .$

Geometric view of the toy optimization problem above. The level curves (curves of constant value) of the objective function are shown. The problem amounts to find the smallest value of $t$ such that $t=f_0(x)$ for some feasible $x$ . The plot also shows the unconstrained minimum of the objective function, located at $\hat{x}=(4,2)$ .

See also:

A toy 2D optimization problem: geometric view via the epigraph form

License

Share This Book