We consider a primal-dual potential reduction algorithm for nonlinear convex optimization problems over symmetric cones. The same comlexity estimates as in the case of the linear objective function are obtained provided a certain nonlinear system of equations can be solved with a given accuracy. This generalizes the result of K. Kortanek,F.Potra and Y.Ye [7]. We further introduce a generalized ...

Throughout this paper, the following conditions are assumed to hold. () n≥  is a positive integer, r ∈ C([t,∞),R+),  < a < b, τ > , σ > ; () p ∈ C([t,∞)× [a,b],R), q ∈ C([t,∞),R+), q ∈ C([t,∞),R+), h ∈ C([t,∞),R); () (u) is a continuously increasing real function with respect to u defined on R, and –(u) satisfies the local Lipschitz condition; () gi ∈ C(R,R), gi(u) satisfy the l...

Let M be a Lagrangian manifold, let the 1-form pdx be globally exact on M and let S(x, p) be defined by dS = pdx on M. Let H(x, p) be convex in p for all x and vanish on M . Let V (x) = inf{S(x, p) : p such that (x, p) ∈ M}. Recent work in the literature has shown that (i) V is a viscosity solution of H(x, ∂V/∂x) = 0 provided V is locally Lipschitz, and (ii) V is locally Lipschitz outside the s...

We introduce a notion of quasi regularity for points with respect to the inclusion F (x) ∈ C, where F is a nonlinear Fréchet differentiable function from Rv to Rm. When C is the set of minimum points of a convex real-valued function h on Rm and F ′ satisfies the L-average Lipschitz condition of Wang, we use the majorizing function technique to establish the semilocal linear/quadratic convergenc...

Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ...

Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ...