Quasi convexity and strong duality

Question

Is there a possibility to prove the strong duality (Lagrangian duality) if the primal problem is quasiconvex? Or is there a technique that proves that the primal problem is convex instead of quasiconvex?

Answer 1

I'm not sure I understand the question clearly enough to give a precise answer, but here is a kind of meta-answer about why you should not expect such a thing to be true.

Generally speaking duality results come from taking some given objective function and adding variable multiples of other functions (somehow connected to constraints) to it. Probably the simplest case is adding various linear functions.

Proving strong duality for a certain type of setup involves understanding the class of functions thus generated, usually using some sort of convexity property. For example, if our original objective function $f$ is convex and we think of adding arbitrary linear functions $L$ to it, the result $f+L$ will always be convex, so we have a good understanding of functions generated in this way, in particular what their optima look like.

Quasiconvexity does not behave nearly so will with respect to the operation of adding linear functions. One way to express this is the following. Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. Then $f+L$ is quasiconvex for all linear maps $L:\mathbb{R}^n\to\mathbb{R}$ if and only if $f$ is convex.

Therefore the class of functions for which the benefits of quasiconvexity (local minima are global, etc.) would help prove strong duality is in some sense just the convex functions. This is not to say strong duality will never hold for quasiconvex objectives, but just that some strong additional conditions are required beyond the usual constraint qualifications used for convex functions: quasiconvexity alone does not buy you much duality-wise.