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Consider the optimization problem \[ \begin{array}{ll} \mbox{minimize} & x_1^2 \\ \mbox{subject to} & x_1 \leq -1, \quad x_1^2 + x_2^2 \leq 2, \end{array} \] with variable $x=(x_1,x_2)$.

The point $(-1,1)$ is a solution.

The optimal value is

The problem is convex.

If an optimization problem is feasible, its optimal value $p^\star$ satisfies $p^\star > -\infty$.

If the optimal value $p^\star$ of an optimization problem satisfies $p^\star < \infty$, then the problem is feasible.

Geometric programming

$f(x,y) = x/y + y/x$ is a posynomial function ($x$ and $y$ are positive variables).

The squareroot of a monomial function is a monomial function.

Suppose $f$, $g$, and $h$ are posynomial functions of a positive variable $z$. The constraint $f(z) + g(z) \leq h(z)$ can be handled by Geometric Programming.

Multi-objective optimization

Suppose $\tilde x$ uniquely minimizes $\max\{f(x),g(x)\}$. Then $\tilde x$ is Pareto optimal for the bi-criterion optimization problem \[ \begin{array}{ll} \mbox{minimize} & (f(x),g(x)). \end{array} \]