Abstract
Some characterizations of solution sets of a convex optimization problem with a convex feasible set described by tangentially convex constraints are given. The results are expressed in terms of convex subdifferentials, tangential subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian-type function and establish a constant pseudo Lagrangian-type property for the solution set. This property is still valid in the case of a pseudoconvex locally Lipschitz objective function, and then used to derive Lagrange multiplier-based characterizations of the solution set. Some examples are given to illustrate the significances of our theoretical results.
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Notes
A tangentially convex function \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) at \(x \in {\mathbb {R}}^n\) is said to be pseudoconvex atx (see [26]) if \( \forall y\in {\mathbb {R}}^n,\; f'(x,y-x)\ge 0 \Longrightarrow f(y)\ge f(x). \)
References
Mangasarian, O.L.: Error bounds for nondegenerate monotone linear complementarity problems. Math. Program. 48, 437–445 (1990)
Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993)
Feltenmark, S., Kiwiel, K.: Dual applications of proximal bundle methods, including Lagrangian relaxation of nonconvex problems. SIAM J. Optim. 10(3), 697–721 (2000)
Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)
Burke, J.V., Ferris, M.: Characterization of solution sets of convex programs. Oper Res Lett 10, 57–60 (1991)
Jeyakumar, V., Yang, X.Q.: Characterizing the solution sets of pseudo-linear 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)
Wu, Z.L., Wu, S.Y.: Characterizations of the solution sets of convex programs and variational inequality problems. J. Optim. Theory Appl. 130, 339–358 (2006)
Yang, X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009)
Castellani, M., Giuli, M.: A characterization of the solution set of pseudoconvex extremum problems. J. Convex Anal. 19, 113–123 (2012)
Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Glob. Optim. 57, 677–693 (2013)
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)
Jeyakumar, V., Lee, G.M., Dinh, N.: Lagrange multiplier conditions characterizing 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 solution sets of mathematical programs in terms of Lagrange multipliers. Optimization 58, 995–1007 (2009)
Zhao, K.Q., Yang, X.M.: Characterizations of the solution set for a class of nonsmooth optimization problems. Optim. Lett. 7, 685–694 (2013)
Mishra, S.K., Upadhyay, B.B., An, L.T.H.: Lagrange multiplier characterizations of solution sets of constrained nonsmooth pseudolinear optimization problems. J. Optim. Theory Appl. 160, 763–777 (2014)
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)
Miao, X.H., Chen, J.S.: Characterizations of solution sets of cone-constrained convex programming problems. Optim. Lett. 9, 1433–1445 (2015)
Lee, G.M., Yao, J.C.: On solution sets for robust optimization problems. J. Nonlinear Convex Anal. 17(5), 957–966 (2016)
Lasserre, J.B.: On representations of the feasible set in convex optimization. Optim. Lett. 4, 1–5 (2010)
Dutta, J., Lalitha, C.S.: Optimality conditions in convex optimization revisited. Optim. Lett. 7(2), 221–229 (2013)
Martinez-Legaz, J.E.: Optimality conditions for pseudoconvex minimization over convex sets defined by tangentially convex constraints. Optim. Lett. 9, 1017–1023 (2015)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Pshenichnyi, B.N.: Necessary Conditions for an Extremum. Marcel Dekker Inc, New York (1971)
Lemaréchal, C.: An introduction to the theory of nonsmooth optimization. Optimization 17(6), 827–858 (1986)
Yamamoto, S., Kuroiwa, D.: Constraint qualifications for KKT optimality condition in convex optimization with locally Lipschitz inequality constraints. Linear Nonlinear Anal. 2(1), 101–111 (2016)
Chieu, N.H., Jeyakumar, V., Li, G., Mohebi, H.: Constraint qualifications for convex optimization without convexity of constraints: new connections and applications to best approximation. Eur. J. Oper. Res. 265(1), 19–25 (2018)
Acknowledgements
The authors would like to express their sincere thanks to anonymous referees for helpful suggestions and valuable comments for the paper. This research was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0026/2555) and the Thailand Research Fund, Grant No. RSA6080077.
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Sisarat, N., Wangkeeree, R. Characterizing the solution set of convex optimization problems without convexity of constraints. Optim Lett 14, 1127–1144 (2020). https://doi.org/10.1007/s11590-019-01397-x
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DOI: https://doi.org/10.1007/s11590-019-01397-x