19,479 research outputs found
Stochastic control with fixed marginal distributions
We briefly describe the so-called Monge-Kantorovich Problem (MKP for short)
which is often referred to as an optimal mass transportation problem and
study the stochastic optimal control problem (SOCP for short)
with fixed initial and terminal distributions.
In particular, we study the so-called Duality Theorem for the SOCP
where continuous semimartingales under consideration have a variable diffusion matrix
and then discuss the relation between the MKP and the SOCP.
We also study the so-called Nelson's Problem, as the SOCP with fixed marginal distributions
at each time, to which we give a new approach from the Duality Theorem.
We finally consider a class of deterministic variational problems with fixed marginal distributions
which is related to the SOCP by extending a class of measures under consideration
Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems
In this paper, a new class of generalized of nonconvex multitime multiobjective variational problems is considered. We prove the sufficient optimality conditions for efficiency and proper efficiency in the considered multitime multiobjective variational problems with univex functionals. Further, for such vector variational problems, various duality results in the sense of Mond-Weir and in the sense of Wolfe are established under univexity. The results established in the paper extend and generalize results existing in the literature for such vector variational problems
Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity
This paper presents a pure complementary energy variational method for
solving anti-plane shear problem in finite elasticity. Based on the canonical
duality-triality theory developed by the author, the nonlinear/nonconex partial
differential equation for the large deformation problem is converted into an
algebraic equation in dual space, which can, in principle, be solved to obtain
a complete set of stress solutions. Therefore, a general analytical solution
form of the deformation is obtained subjected to a compatibility condition.
Applications are illustrated by examples with both convex and nonconvex stored
strain energies governed by quadratic-exponential and power-law material
models, respectively. Results show that the nonconvex variational problem could
have multiple solutions at each material point, the complementary gap function
and the triality theory can be used to identify both global and local extremal
solutions, while the popular (poly-, quasi-, and rank-one) convexities provide
only local minimal criteria, the Legendre-Hadamard condition does not guarantee
uniqueness of solutions. This paper demonstrates again that the pure
complementary energy principle and the triality theory play important roles in
finite deformation theory and nonconvex analysis.Comment: 23 pages, 4 figures. Mathematics and Mechanics of Solids, 201
A Duality Approach to Error Estimation for Variational Inequalities
Motivated by problems in contact mechanics, we propose a duality approach for
computing approximations and associated a posteriori error bounds to solutions
of variational inequalities of the first kind. The proposed approach improves
upon existing methods introduced in the context of the reduced basis method in
two ways. First, it provides sharp a posteriori error bounds which mimic the
rate of convergence of the RB approximation. Second, it enables a full
offline-online computational decomposition in which the online cost is
completely independent of the dimension of the original (high-dimensional)
problem. Numerical results comparing the performance of the proposed and
existing approaches illustrate the superiority of the duality approach in cases
where the dimension of the full problem is high.Comment: 21 pages, 8 figure
Noether's symmetry theorem for nabla problems of the calculus of variations
We prove a Noether-type symmetry theorem and a DuBois-Reymond necessary
optimality condition for nabla problems of the calculus of variations on time
scales.Comment: Submitted 20/Oct/2009; Revised 27/Jan/2010; Accepted 28/July/2010;
for publication in Applied Mathematics Letter
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