On the Lower Semicontinuity of the Value Function and Existence of Solutions in Quasiconvex Optimization

This paper provides sufficient conditions ensuring the lower semicontinuity of the value function $$\begin{aligned} \psi (a):= \inf \Big \{f(x):g_1(x)\le a_1,\ldots ,g_m(x)\le a_m\Big \},\quad a=(a_1,\ldots ,a_m), \end{aligned}$$ ψ ( a ) : = inf { f ( x ) : g 1 ( x ) ≤ a 1 , … , g m ( x ) ≤ a m } , a = ( a 1 , … , a m ) , at 0, under quasiconvexity assumptions on f and $$g_i$$ g i , although there are results where convexity of some $$g_i$$ g i will be required. In some situations, our conditions will imply also the existence of points where the value $$\psi (0)$$ ψ ( 0 ) is achieved. In convex optimization, it is known that zero duality gap is equivalent to the lower semicontinuity of $$\psi $$ ψ at 0. Here, the dual problem is defined in terms of the linear Lagrangian. We recall that convexity of the closure of the set $$(f,g_1,\ldots ,g_m)({\mathbb {R}}^n)+{\mathbb {R}}_+^{1+m}$$ ( f , g 1 , … , g m ) ( R n ) + R + 1 + m and lower semicontinuity of $$\psi $$ ψ at 0 imply zero duality gap. In addition, our results provide much more information than those existing in the literature. Several examples showing the applicability of our approach and the non applicability of any other result elsewhere are exhibited. Furthermore, we identify a suitable large class of functions (quadratic linear fractional) to which f and $$g_i$$ g i could belong to and our results apply.