By Brinkhuis J., Zhang S.
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Extra resources for A D-induced duality and its applications
Optimization: insights and applications. Princeton University Press, Princeton (2005) 3. : Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18, 1035–1064 (1997) 4. : Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1993) 5. : Duality results for conic convex programming. Technical Report 9719/A, Econometric Institute, Erasmus University Rotterdam, The Netherlands (1997) 6. : Multivariate nonnegative quadratic mappings. SIAM J.
This is equivalent to requiring that the epigraph of φ, that is, epi(φ) = φ(u) + r u r∈ +, u∈U , is a convex cone. We note that the epigraph of a sublinear function φ on X is a closed, solid, pointed convex cone. A sublinear function φ on U is positive if φ(u) > 0 for all nonzero u. We call two convex cones E ⊆ Y and F ⊆ Z equivalent if there exists a linear bijective transformation Y → Z which maps E bijectively onto F. We say that a convex cone E ⊆ Y is represented by a sublinear mapping φ on U if the convex cones E ⊆ Y and epi(φ) ⊆ × U are equivalent.
As immediate research topics we pose the following two questions: (1) Is it possible to describe using LMI’s the cone (U)∗D where U is a second order cone in X and D is a second order cone in W? As we remarked in Sect. 4, this amounts to the LMI representation of the bi-positive set B(C, D) where C and D are second order cones. (2) In Sect. 5 we showed that the dual of a multi-objective conic optimization problem, denoted by (D)D , always has an attainable global optimal solution under some suitable Slater type regularity conditions, provided that the multi-objectives are ordered by the first orthant.