Taiwanese Journal of Mathematics

Stable Conical Regularization by Constructible Dilating Cones with an Application to $L^{p}$-constrained Optimization Problems

Baasansuren Jadamba, Akhtar A. Khan, and Miguel Sama

Full-text: Open access


We study a convex constrained optimization problem that suffers from the lack of Slater-type constraint qualification. By employing a constructible representation of the constraint cone, we devise a new family of dilating cones and use it to introduce a family of regularized problems. We establish novel stability estimates for the regularized problems in terms of the regularization parameter. To show the feasibility and efficiency of the proposed framework, we present applications to some $L^{p}$-constrained least-squares problems.

Article information

Taiwanese J. Math., Volume 23, Number 4 (2019), 1001-1023.

Received: 10 January 2018
Revised: 23 July 2018
Accepted: 1 November 2018
First available in Project Euclid: 18 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90C20: Quadratic programming 90C31: Sensitivity, stability, parametric optimization 90C46: Optimality conditions, duality [See also 49N15]

perturbation theory convex optimization half-space representation conical regularization dilating cones


Jadamba, Baasansuren; Khan, Akhtar A.; Sama, Miguel. Stable Conical Regularization by Constructible Dilating Cones with an Application to $L^{p}$-constrained Optimization Problems. Taiwanese J. Math. 23 (2019), no. 4, 1001--1023. doi:10.11650/tjm/181103. https://projecteuclid.org/euclid.twjm/1563436878

Export citation


  • W. Alt, Stability of solutions for a class of nonlinear cone constrained optimization problems I: Basic theory, Numer. Funct. Anal. Optim. 10 (1989), no. 11-12, 1053–1064.
  • D. Azagra and J. Ferrera, Every closed convex set is the set of minimizers of some $C^{\infty}$-smooth convex function, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3687–3692.
  • J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000.
  • J. M. Borwein and J. D. Vanderwerff, Constructible convex sets, Set-Valued Anal. 12 (2004), no. 1-2, 61–77.
  • J. M. Borwein and D. Zhuang, Super efficiency in vector optimization, Trans. Amer. Math. Soc. 338 (1993), no. 1, 105–122.
  • O. Davydov, Approximation by piecewise constants on convex partitions, J. Approx. Theory 164 (2012), no. 2, 346–352.
  • A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 17, Springer-Verlag, New York, 2003.
  • M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, Version 2.1. http://cvxr.com/cvx
  • M. D. Gunzburger, Perspectives in Flow Control and Optimization, Advances in Design and Control 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.
  • M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl. 36 (1982), no. 3, 387–407.
  • M. Hohenwarter, GeoGebra: Ein Softwaresystem für dynamische Geometrie und Algebra der Ebene, Paris Lodron University, Salzburg, Austria, 2002.
  • V. Isakov, Inverse Problems for Partial Differential Equations, Second edition, Applied Mathematical Sciences 127, Springer, New York, 2006.
  • K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control 15, Society of Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.
  • B. Jadamba, A. A. Khan and M. Sama, Regularization for state constrained optimal control problems by half spaces based decoupling, Systems Control Lett. 61 (2012), no. 6, 707–713.
  • ––––, Error estimates for integral constraint regularization of state-constrained elliptic control problems, Comput. Optim. Appl. 67 (2017), no. 1, 39–71.
  • B. Jiménez, V. Novo and M. Sama, An extension of the basic constraint qualification to nonconvex vector optimization problems, J. Global Optim. 56 (2013), no. 4, 1755–1771.
  • A. A. Khan and M. Sama, A new conical regularization for some optimization and optimal control problems: convergence analysis and finite element discretization, Numer. Funct. Anal. Optim. 34 (2013), no. 8, 861–895.
  • A. A. Khan, C. Tammer and C. Zălinescu, Set-valued Optimization: An Introduction with Applications, Vector Optimization, Springer, Heidelberg, 2015.
  • P. Kogut, G. Leugering and R. Schiel, On Henig regularization of material design problems for quasi-linear $p$-biharmonic equation, Appl. Math. 7 (2016), no. 14, 1547–1570.
  • ––––, On the relaxation of state-constrained linear control problems via Henig dilating cones, Control Cybernet. 45 (2016), no. 2, 131–162.
  • I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I: Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications 74, Cambridge University Press, Cambridge, 2000.
  • E. K. Makarov and N. N. Rachkovski, Unified representation of proper efficiency by means of dilating cones, J. Optim. Theory Appl. 101 (1999), no. 1, 141–165.
  • K. Martin, C. T. Ryan and M. Stern, The Slater conundrum: duality and pricing in infinite-dimensional optimization, SIAM J. Optim. 26 (2016), no. 1, 111–138.
  • A. Mas-Colell and W. R. Zame, Equilibrium theory in infinite-dimensional spaces, in: Handbook of Mathematical Economics IV, 1835–1898, Handbooks in Econom. 1, North-Holland, Amsterdam, 1991.
  • R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics 183, Springer-Verlag, New York, 1998.