Abstract
In this paper, we study some problems with a continuously differentiable and quasiconvex objective function. We prove that exactly one of the following two alternatives holds: (I) the gradient of the objective function is different from zero over the solution set, and the normalized gradient is constant over it; (II) the gradient of the objective function is equal to zero over the solution set. As a consequence, we obtain characterizations of the solution set of a program with a continuously differentiable and quasiconvex objective function, provided that one of the solutions is known. We also derive Lagrange multiplier characterizations of the solutions set of an inequality constrained problem with continuously differentiable objective function and differentiable constraints, which are all quasiconvex on some convex set, not necessarily open. We compare our results with the previous ones. Several examples are provided.
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References
Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)
Burke, J.V., Ferris, M.C.: Characterization of the solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)
Jeyakumar, V., Yang, X.Q.: On characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl. 87, 747–755 (1995)
Ivanov, V.I.: First-order characterizations of pseudoconvex functions. Serdica Math. J. 27, 203–218 (2001)
Ivanov, V.I.: Characterizations of the solution sets of generalized convex minimization problems. Serdica Math. J. 29, 1–10 (2003)
Ivanov, V.I.: Optimality conditions and characterizations of the solution sets in generalized convex problems and variational inequalities. J. Optim. Theory Appl. 158, 65–84 (2013)
Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Global Optim. 57, 677–693 (2013)
Wu, Z.: The convexity of the solution set of a pseudoconvex inequality. Nonlinear Anal. TMA 69, 1666–1674 (2008)
Yang, X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009)
Liu, C., Yang, X., Lee, H.: Characterizations of the solution sets of pseudoinvex programs and variational inequalities. J. Inequal. Appl. 2011, 32 (2011)
Ivanov, V.I.: Higher order invex functions and higher order pseudoinvex ones. Appl. Anal. 92, 2152–2167 (2013)
Smietanski, M.: A note on characterization of solution sets of pseudolinear programming problems. Appl. Anal. 91, 2095–2104 (2012)
Barani, A.: Convexity of the solution set of a pseudoconvex inequality in Riemannian manifolds. Numer. Funct. Anal. Optim. 39, 588–599 (2018)
Jeyakumar, V., Lee, G.M., Dinh, N.: Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs. J. Optim. Theory Appl. 123, 83–103 (2004)
Jeyakumar, V., Lee, G.M., Dinh, N.: Characterizations of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174, 1380–1395 (2006)
Dinh, N., Jeyakumar, V., Lee, G.M.: Lagrange multiplier characterizations of solution sets of constrained pseudolinear optimization problems. Optimization 55, 241–250 (2006)
Son, T.Q., Dinh, N.: Characterizations of optimal solution sets of convex infinite programs. Top 16, 147–163 (2008)
Lalitha, C.S., Mehta, M.: Characterizations of the solution sets of mathematical programs in terms of Lagrange multipliers. Optimization 58, 995–1007 (2009)
Zhao, K.Q., Yang, X.M.: Characterizations of solution set for a class of nonsmooth optimization problems. Optim. Lett. 7, 685–694 (2013)
Son, T.Q., Kim, D.S.: A new approach to characterize the solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261, 333–340 (2014)
Long, X.J., Peng, Z.Y., Wang, X.: Characterizations of the solution set for nonconvex semi-infinite programming problems. J. Nonlinear Convex Anal. 17, 251–265 (2016)
Penot, J.-P.: Characterization of solution set of quasiconvex programs. J. Optim. Theory Appl. 117, 627–636 (2003)
Ivanov, V.I.: Optimization and variational inequalities with pseudoconvex functions. J. Optim. Theory Appl. 146, 602–616 (2010)
Suzuki, S., Kuroiwa, D.: Characterizations of solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. J. Global Optim. 62, 431–441 (2015)
Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for non-essentially quasiconvex programming. Optim. Lett. 11, 1699–1712 (2017)
Mangasarian, O.L.: Nonlinear Programming. Classics in Applied Mathematics. SIAM, Philadelphia (1994)
Arrow, K.J., Enthoven, A.C.: Quasi-concave programming. Econometrica 29, 779–800 (1961)
Gordan, P.: Über die Auflösungen linearer Gleichungen mit reelen coefficienten. Math. Ann. 6, 23–28 (1873)
Bertsekas, D.P., Nedic, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)
Avriel, M., Diewert, W., Schaible, S., Zang, I.: Generalized Concavity, Classics in Applied Mathematics (Originally published: New York (1988)), vol. 63. SIAM, Philadelphia (2010)
Greenberg, H.J., Pierskalla, W.P.: Quasiconjugate functions and surrogate duality. Cah. Cent. Etud. Rech. Oper. 15, 437–448 (1973)
Martinez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19, 603–652 (1988)
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Vaithilingam Jeyakumar.
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Ivanov, V.I. Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems. J Optim Theory Appl 181, 144–162 (2019). https://doi.org/10.1007/s10957-018-1379-1
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DOI: https://doi.org/10.1007/s10957-018-1379-1
Keywords
- Quasiconvex function
- Characterizations of the solution set
- Quasiconvex program
- Pseudoconvex function
- KKT Conditions