Global optimization by canonical dual function

In this paper, the canonical dual function (Gao, 2004 [4]) is used to solve a global optimization. We find global minimizers by backward differential flows. The backward flow is created by the local solution to the initial value problem of an ordinary differential equation. Some examples and applications are presented.     

Properties of the Lagrange multipliers in the Pontryagin maximum principle for optimal control problems with state constraints

We study the Pontryagin maximum principle for an optimal control problem with state constraints. We analyze the continuity of a vector function µ (which is one of the Lagrange multipliers corresponding to an extremal by virtue of the maximum principle) at the points where the extremal trajectory meets the boundary of the set given by the state constraints. We obtain sufficient conditions for the continuity of µ in terms of the smoothness of the extremal trajectory.     

A Finite Horizon Linear Quadratic Optimal Stochastic Control Problem Driven by Both Brownian Motion and L\'{e}vy Processes

We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

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??We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

Moving Object in ℝ<Superscript>2</Superscript> and a Group of Observers

We formulate an extremal problem of constructing a trajectory of a moving object that is farthest from a group of observers with fixed visibility cones. Under some constraints on the arrangement of the observers, we give a characterization and a method of construction of an optimal trajectory.     

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.     

广义LQ最优控制问题的极小化序列

将经典LQ问题的评价泛函中关于控制变量的二次型推广为一类偶次多项式,证明了这类广义LQ无约束最优控制问题的一个等价扩张逼近可由一列半径递增的球约束最优控制问题加以实现.进而利用P0ntryagin极值原理建立相应的球约束最优控制问题的二次规划,并通过Canonical倒向微分流及不动点定理,求解常微分方程边值问题,得到球约束最优控制问题的最优值.随着约束球半径趋于无穷大,形成原广义LQ最优控制问题的一个极小化序列,从而得到原问题的最优值.     

Solution of the unconditional extremum problem for a linear-fractional integral functional on a set of probability measures

A new, generalized and strengthened, form of an assertion about an extremum of a linear-fractional integral functional given on a set of probability measures is presented. It is shown that the solution of the extremal problem for such a functional is completely determined by the extremal properties of the so-called test function, which is the ratio of the integrands of the numerator and the denominator. On the basis of this assertion, a theorem on an optimal strategy for controlling a semi-Markov process with a finite set of states is proved. In particular, it is established that if the test function of the objective functional of a control problem attains a global extremum, then an optimal control strategy exists, is deterministic, and is determined by the point of global extremum. The corresponding assertions are also obtained for the case where the test function does not attain the global extremum.     

Characterization of the Value Process in Robust Efficient Hedging

This paper examines the value function of a partial hedging problem under model ambiguity. The study is based on a dual representation of the value function obtained by the authors. We formulate a family of control problems, whose value processes are characterized as solutions of a backward stochastic differential equation and give a sufficient condition to identify optimal controls.     

On the generalized Ritt problem as a computational problem

The Ritt problem asks if there is an algorithm that decides whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing whether a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of this equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking.