A D-induced duality and its applications by Brinkhuis J., Zhang S.

By Brinkhuis J., Zhang S.

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Math. Oper. Res. 23, 769–805 (1998) 2. : 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.

Approximating quadratic programming with bound and quadratic constraints. Math. Progr. 84, 219–226 (1999) 14. : A new self-dual embedding method for convex programming. J. Global Optim.

Now we can formulate and so carry out the second step as follows. 2 Each closed, solid, pointed convex cone C can be represented by a positive sublinear function φ. Proof One has C∗ = {0}, by separation of convex sets, and so one can choose a nonzero element y ∈ C∗ . Let U = {x ∈ X : x, y = 0} and define the function φ on U by φ(u) = inf{ρ | u + ρy ∈ C}. It is readily verified that φ is a positive, sublinear function that represents C. For instance, if C is a second order cone, then the corresponding sublinear function can be simply the Euclidean norm φ(u) = u ; if C is an orthant, then φ(u) = max{maxi ui , 0}; if C is the cone of positive semidefinite matrices, then φ(u) = max{λmax (Mat(u)), 0}, where ‘Mat(u)’ stands for the stacking operation A D-induced duality and its applications 179 to create a symmetric matrix from u, and ‘λmax (X)’ stands for the maximum eigenvalue of X.

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