Convex optimization number of solutions

  • Does convex optimization have a unique solution?

    Convex optimization problems are attractive because they always have a unique solution..

In fact a convex optimization problem may have 0, 1 or uncountably infinite solutions. Xopt is an empty set when no solutions are obtained (e.g. in a minimization of a linear function). Exactly 1 solution is obtained when the criterion f is strictly convex (e.g. when f(x) = x2).
In fact a convex optimization problem may have 0, 1 or uncountably infinite solutions. Xopt is an empty set when no solutions are obtained (e.g. in a minimization of a linear function). Exactly 1 solution is obtained when the criterion f is strictly convex (e.g. when f(x) = x2).

Is convex optimization a generalization of linear programming?

Comparing (1 3) and (1

2), we see that convexity is more general than linearity: inequality replaces the more restrictive equality, and the inequality must hold only for certain values of αand β

Since any linear program is therefore a convex optimization problem, we can consider convex optimization to be a generalization of linear programming

Which stochastic optimization problem is convex?

1; : : : ; m; where the expectation is with respect to u

Show that if fi are convex in x for each u, then this stochastic optimization problem is convex

Worst-case optimization

We consider the problem Show that if fi are convex in x for each u, then this worst-case optimization problem is convex

Finite set of possible parameter values


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