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In a recent problem set for my optimization course, following problem was mentioned.

Problem

Consider constrained minimization problem of the following form: $$\min_{\textbf{x}\in \Omega} f(\textbf{x})$$

Show that if $f$ is convex and the feasible region is convex, then the local solutions of the problems are also global solutions. Also show that the set of global solutions is convex. A problem with convex objective function and convex feasible region is said to be convex optimization problem.

My attempt

I considered $\textbf{x}^\star$ as the local minimizer. I tried solving the problem with contradiction.

Since its mentioned that the feasible solution is a convex set, I considered a point $\tilde{\textbf{x}}$ which is in the set such that, $$f(\tilde{\textbf{x}}) \leq f({{\textbf{x}^\star}})$$

I would proceed like I would in the unconstrained optimization but how do I incorporate information about the feasible set $\Omega$.

Where feasible set is defined as $$\Omega = \{x|c_{i}(x) = 0 \text{ }\forall\ i \in \mathcal{E} ; c_{i}(x) \geq 0 \text{ }\forall\ i \in \mathcal{I} \}$$

where, $c_{i}(x)$ are the constraints, $\mathcal{E}$ is the set of equality constrains while $\mathcal{I}$ is the set of inequality constrains

Nirav L Lekinwala
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1 Answers1

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Let $x^{*} \in \mathbb{R}^{n}$ be a local minizer of $f$ on $\Omega$. Thus there exists an $\epsilon>0$ such that $$f(x)\geq f(x^{*})$$$$x,x^{*}\in\Omega$$ $$||x-x^{*}||<\epsilon$$

Let $y\in\Omega$ and choose $\lambda\in (0,1)$. Then by convexity of $\Omega$, $x^{*}+\lambda(y-x^{*})=(1-\lambda)x^{*}+\lambda y\in\Omega$.

By convexity of $f$, $$f(x^{*})\leq f(\left(1-\lambda)x^{*}+\lambda(y)\right)\leq(1-\lambda) f(x^{*})+\lambda f(y).$$

Rewriting the above inequality we get $$f(x^{*})\leq f(x^{*})-\lambda f(x^{*})+\lambda f(y).$$ Rearranging terms, we arrive at the following result: $$f(x^{*})\leq f(y) \quad \forall y\in\Omega.$$

Thus, $x^{*}$ is a global minimizer of $f$ on $\Omega$.

To show the set of global minimizers is convex can be easily done by using the basic definition of a convex set. The set-up of the problem is as follows. Let $G:=\{x^{*}| f(x^{*})\leq f(x) \quad \forall x\in\Omega\}$. Show that for all $x^{*}_{1}, x^{*}_2 \in G$, $\lambda (x^{*}_{1})+(1-\lambda) (x^{*}_{2})\in G,\quad \lambda\in [0,1]$.

Andrew Gross
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