Definition
Convex programming problem is the problem that:
- The feasible region is a convex set
- If the goal is minimizing, the objective function is convex over the feasible region
- If the goal is maximizing, the objective function is concave over the feasible region
For a problem:
minimize F(x)
subject to:
The following condition are sufficient to guarantee the feasible region is a convex set:
is concave for j = 1, 2, ... , t
is linear for j = t+1, t+2, ... , m
strictly convex program is the convex program with a strict objective function with respect to all problem variables.
Properties
Properties of convex programming:
- All local optima are global optima, but they may not be unique
- Sometimes convex programming problems have exactly one optimal solution (unique solution).
- For strictly convex program, any locally optimal solution is the unique solution