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The purpose of this paper is to present an alternative proof of the existence of the Walrasian equilibrium for the Arrow-Debreu-McKenzie model by the variational inequality technique. Moreover, examples of the generalized Arrow-Debreu-McKenzie model are given in which the price vector can reach the boundary of the orthant allowing a commodity to be of price zero at equilibrium. In such a case its supply exceeds demand. It is worth mentioning that utility functions in this model are allowed not to...
The finite element approximation of optimal control problems for
semilinear elliptic partial differential equation is considered,
where the control belongs to a finite-dimensional set and state
constraints are given in finitely many points of the domain. Under
the standard linear independency condition on the active gradients
and a strong second-order sufficient optimality condition, optimal
error estimates are derived for locally optimal controls.
The goal of this paper is to derive some error estimates for the numerical discretization of some optimal control problems governed by semilinear elliptic equations with bound constraints on the control and a finitely number of equality and inequality state constraints. We prove some error estimates for the optimal controls in the norm and we also obtain error estimates for the Lagrange multipliers associated to the state constraints as well as for the optimal states and optimal adjoint states....
The goal of this paper is to derive some error estimates for the
numerical discretization of some optimal control problems governed
by semilinear elliptic equations with bound constraints on the
control and a finitely number of equality and inequality state
constraints. We prove some error estimates for the optimal
controls in the L∞ norm and we also obtain error estimates
for the Lagrange multipliers associated to the state constraints
as well as for the optimal states and optimal adjoint states.
...
In this paper, a distributed optimal consensus problem is investigated to achieve the optimization of the sum of local cost function for a group of agents in the Euler-Lagrangian (EL) system form. We consider that the local cost function of each agent is only known by itself and cannot be shared with others, which brings challenges in this distributed optimization problem. A novel gradient-based distributed continuous-time algorithm with the parameters of EL system is proposed, which takes the distributed...
The paper presents the Monotone Structural Evolution, a direct computational method of optimal control. Its distinctive feature is that the decision space undergoes gradual evolution in the course of optimization, with changing the control parameterization and the number of decision variables. These structural changes are based on an analysis of discrepancy between the current approximation of an optimal solution and the Maximum Principle conditions. Two particular implementations, with spike and...
Here we present an approximation method for a rather broad class of first order
variational problems in spaces of piece-wise constant functions over
triangulations of the base domain. The convergence of the method is based on an
inequality involving norms obtained by Nečas and on the general
framework of Γ-convergence theory.
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