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

Author

Listed:
  • Fabián Flores-Bazán

    (Universidad de Concepción)

  • Filip Thiele

    (Universidad de Concepción)

Abstract

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.

Suggested Citation

  • Fabián Flores-Bazán & Filip Thiele, 2022. "On the Lower Semicontinuity of the Value Function and Existence of Solutions in Quasiconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 390-417, November.
  • Handle: RePEc:spr:joptap:v:195:y:2022:i:2:d:10.1007_s10957-022-02079-y
    DOI: 10.1007/s10957-022-02079-y
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    References listed on IDEAS

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    1. Fabián Flores-Bazán & William Echegaray & Fernando Flores-Bazán & Eladio Ocaña, 2017. "Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap," Journal of Global Optimization, Springer, vol. 69(4), pages 823-845, December.
    2. Alberto Cambini & Laura Martein, 2009. "Generalized Convexity and Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-70876-6, July.
    3. Emil Ernst & Michel Volle, 2013. "Zero Duality Gap for Convex Programs: A Generalization of the Clark–Duffin Theorem," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 668-686, September.
    4. Fabián Flores-Bazán & Fernando Flores-Bazán & Cristián Vera, 2015. "Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives," Journal of Global Optimization, Springer, vol. 63(1), pages 99-123, September.
    5. Radu Ioan Bot, 2010. "Conjugate Duality in Convex Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-04900-2, July.
    6. A. E. Ozdaglar & P. Tseng, 2006. "Existence of Global Minima for Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 128(3), pages 523-546, March.
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