[PDF] convex optimization unique solution

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  • Does convex optimization 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).

  • How do you solve convex optimization?

    Unconstrained convex optimization can be easily solved with gradient descent (a special case of steepest descent) or Newton's method, combined with line search for an appropriate step size; these can be mathematically proven to converge quickly, especially the latter method.

  • What is the unique minimum of a strictly convex function?

    If f is strictly convex, then there exists at most one local minimum of f in X.
    Consequently, if it exists it is the unique global minimum of f in X.

  • What is the unique minimum of a strictly convex function?

    In a convex optimization problem, the feasible region -- the intersection of convex constraint functions -- is a convex region, as pictured below. With a convex objective and a convex feasible region, there can be only one optimal solution, which is globally optimal.

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