Determination of optimal feedback terminal controllers for general boundary conditions using generating functions

Given a nonlinear system and a performance index to be minimized, we present a general approach to expressing the finite time optimal feedback control law applicable to different types of boundary conditions. Starting from the necessary conditions for optimality represented by a Hamiltonian system, we solve the Hamilton-Jacobi equation for a generating function for a specific canonical transformation. This enables us to obtain the optimal feedback control for fundamentally different sets of boundary conditions only using a series of algebraic manipulations and partial differentiations. Furthermore, the proposed approach reveals an insight that the optimal cost functions for a given dynamical system can be decomposed into a single generating function that is only a function of the dynamics plus a term representing the boundary conditions. This result is formalized as a theorem. The whole procedure provides an advantage over methods rooted in dynamic programming, which require one to solve the Hamilton-Jacobi-Bellman equation repetitively for each type of boundary condition. The cost of this favorable versatility is doubling the dimension of the partial differential equation to be solved.     

无限维的不可微凸规划

本文在局部凸Ｈａｕｓｄｏｒｆｆ线性拓扑空间中讨论了一类不可微的凸规划，得到了Ｋｕｈｎ－Ｔｕｃｋｅｒ最优性条件和对偶定理。     

Greedily computing associative aggregations on sliding windows

We present an algorithm for combining the elements of subsequences of a sequence with an associative operator. The subsequences are given by a sliding window of varying size. Our algorithm is greedy and computes the result with the minimal number of operator applications.     

The solution and duality of imprecise network problems

Duality properties have been investigated by many researchers in the recent literature. They are introduced in this paper for a fully fuzzified version of the minimal cost flow problem, which is a basic model in network flow theory. This model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes. First, we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper. By applying Hukuhara’s difference, approximated and exact multiplication and Wu’s scalar production, we exhibit the flow in network models. Then, we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified MCFP to acquire fuzzy optimal flows. To give duality theorems, we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms. Finally, we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision-making problems.     

Theory and methodology on the global optimal solution to a General Reverse Logistics Inventory Model for deteriorating items and time-varying rates

In this paper, we present a unified general inventory model for integrated production of new items and remanufacturing of returned items for an infinite planning horizon. Our model considers a production environment that consists of three shops. The first shop is for remanufacturing returned items, the second shop is for manufacturing new items, while the third shop is for collecting returned items to be remanufactured in the first shop. The system is subject to joint production and remanufacturing options, the first one is to produce new items while the second one is to reproduce/recycle the returned items “as-good-as new”. Items deteriorate while they are in storage, and production, remanufacturing, demand, return, and deterioration rates are arbitrary functions of time. A closed form for the total relevant costs as well as a rigorous mathematical proof, which shows the global optimality of the solution to the underlying inventory system are introduced. Illustrative examples, which explain the application of the theoretical results as well as their numerical verifications, are also given.     

Optimality and stability of the K-hyperline clustering algorithm

K-hyperline clustering is an iterative algorithm based on singular value decomposition and it has been successfully used in sparse component analysis. In this paper, we prove that the algorithm converges to a locally optimal solution for a given set of training data, based on Lloyd’s optimality conditions. Furthermore, the local optimality is shown by developing an Expectation-Maximization procedure for learning dictionaries to be used in sparse representations and by deriving the clustering algorithm as its special case. The cluster centroids obtained from the algorithm are proved to tessellate the space into convex Voronoi regions. The stability of clustering is shown by posing the problem as an empirical risk minimization procedure over a function class. It is proved that, under certain conditions, the cluster centroids learned from two sets of i.i.d. training samples drawn from the same probability space become arbitrarily close to each other, as the number of training samples increase asymptotically.     

DCAF: An MPEG-21 Dynamic Content Adaptation Framework

Universal Multimedia Access aims at providing a gratifying end user-experience by either adapting the content, be it static or dynamic, to suit the usage environment or adapting the usage environment, be it client- or server-centric, to suit content. This paper presents our MPEG-21 Dynamic Content Adaptation Framework, acronym DCAF, which uses a fusion of Genetic Algorithms and Strength Pareto Optimality to adapt content in order to suit the usage environment.

Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.