Polynomial approximation based learning search

Two stage stochastic programming is an important part in the whole area of stochastic programming, and is widely spread in multiple disciplines, such as financial management, risk management, and logistics. The two stage stochastic programming is a natural extension of linear programming by incorporating uncertainty into the model. This thesis solves the two stage stochastic programming using a novel approach. For most two stage stochastic programming model instances, both the objective function and constraints are convex but non-differentiable, e.g. piecewise-linear, and thereby solved by the first gradient-type methods. When encountering large scale problems, the performance of known methods, such as the stochastic decomposition (SD) and stochastic approximation (SA), is poor in practice. This thesis replaces the objective function and constraints with their polynomial approximations. That is because polynomial counterpart has the following benefits: first, the polynomial approximation will preserve the convexity; Second, the polynomial approximation will uniformly converge to the original objective/constraints with arbitrary accuracy; and third, the polynomial approximation will not only provide good estimation on the original objectives/functions but also their gradients/sub-gradients. All these features enable us to apply convex optimization techniques for large scale problems. Hence, the thesis applies SAA, polynomial approximation method and then steepest descent method in combination to solve the large-scale problems effectively and efficiently.

International audience

Groundwater is being contaminated rapidly due to various anthropogenic activities and geogenic sources. In this direction, assessment of water quality analysis is the basic requirement for nurturing the human being and its evolution. The Water Quality Index (WQI) parameter has been widely used in determining water quality globally. The study aims to provide the suitability of groundwater in the specified region using the polynomial approximation method for drinking and irrigation purposes along with the computation of WQI using the conventional method. Weierstrass's polynomial approximation theorem along with longitudinal and latitudinal values has been used to evaluate the polynomial regarding various physicochemical parameters. To validate the obtained results from the present approach, groundwater quality data collected and analyzed from the Pindrawan tank area in Raipur district, Chhattisgarh, India, have been used. The result is obtained, i.e., the intermediate value of the parameters obtained correctly from the mathematical modeling, with an average error of 7%. This polynomial approximation method can also be used as the substitute of inverse modeling to determine the location of the source in the two-dimensional system. The approach output can be beneficial to administrators in making decisions on groundwater quality and gaining insight into the tradeoff between system benefit and environmental requirement.

The small-signal security region (SSSR) is an important concept in the security analysis of power systems, especially for the problems of low-frequency oscillation. In this paper, a polynomial approximation method is proposed to construct an analytical expression of the boundaries of the damping-ratio-based small-signal security region (D-SSSR) by using the implicit function approach. Compared with the conventional small-signal security region based on the Hopf bifurcation, our method is more practical and suitable for the dispatching of power systems, since it directly depicts the weak damped boundaries in the high-dimensional parameter space. The formulations of the linear and quadratic approximation are presented in detail. The effectiveness and accuracy of the algorithm is verified through case studies on a rudimentary power system.

NP-hard cases of the single-item capacitated lot-sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known that produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is first developed for a quite general model, which has concave backlogging and production cost functions and arbitrary (monotone) holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models.

A general approach to approximation problems of the Bernstein type

Coil motion noise is the main noise source in AEM data caused by the movement of the receiver coil in the Earth’s magnetic field. The characteristics of both time domain and frequency domain of motion noise are analyzed. In this paper we present a scheme of removing motion noise which employs polynomial to represent motion noise for each half cycle. A set of equations are set up according to the characteristics of AEM data, and Lagrange method is employed to solve minimum problem with two constraint conditions together. The polynomial coefficients are calculated by matrix inverse and finally motion noise is subtracted from the AEM data half cycle by half cycle. Polynomial approximation method is applied to the synthetic data contaminated by low frequency motion noise comparing with traditional high pass filter method. The corrected AEM data shows polynomial approximation method is more effective in dealing with motion noise.

In Tl2SO2+Na2SO4+di(2‐ethylhexyl)dithiophosphoric acid+n‐C8H18+water system, the Tl+ equilibrium molalities were measured at ionic strength from 0.1 to 2.0 mol·kg‐1 containing Na2SO4 as supporting electrolyte in aqueous phase and at constant molality of extractant at temperatures from 278.15 to 303.15 K in organic phase. The standard extraction constants K0 at various temperatures were obtained by methods of extrapolation and polynomial approximation. Thermodynamic quantities for the extraction process were calculated.

To overcome the problem of determining the number of hidden-layer neurons in feed-forward neural networks, a polynomial feed-forward neural network with a single hidden layer is presented based on the theory of polynomial approximation, where the polynomials are employed as the activation functions of hidden-layer neurons, and the weights between input layer and hidden layer are set to be 1. We only need to adjust the weights between hidden layer and output layer. Then, using the least square method, we could deduce the formula of computing weights directly. Furthermore, the basic ideas of the sieve-decrease algorithm of polynomial neural networks are described and discussed in details, together with several new concepts, such as weight-sieve, sieve-pore diameter, sieve-decrease rate,etc. Two illustrative computer-simulations substantiate that the improved polynomial feed-forward neural networks possess superior performance, and show that the number of hidden neurons decreases respectively by 98.19% and 80%, as compared to primal neural networks.

Mobile robot navigation is to find an optimal path to guide the movement of the robot, so path planning is guaranteed to find a feasible optimal path. However, the path planning problem must be solve two problems, i.e., the path must be kept away from obstacles or avoid the collision with obstacles and the length of path should be minimized. In this paper, a path planning algorithm based on random coding particle swarm optimization (RCPSO) algorithm is proposed to get the optimal collision-free path. Dijstra algorithm is applied to search a sub-optimal collision-free path in our algorithm; then the RCPSO algorithm is developed to tackle this optimal path planning problem in order to generate the global optimal path. The crossover operator of genetic algorithm and random coding are introduced into the particle swarm optimization to optimize the location of the sub-optimal path. The experiment results show that the proposed method is effective and feasible compared with different algorithms.

When reinforcement learning with a deep neural network is applied to heuristic search, the search becomes a learning search. In a learning search system, there are two key components: (1) a deep neural network with sufficient expression ability as a heuristic function approximator that estimates the distance from any state to a goal; (2) a strategy to guide the interaction of an agent with its environment to obtain more efficient simulated experience to update the Q-value or V-value function of reinforcement learning. To date, neither component has been sufficiently discussed. This study theoretically discusses the size of a deep neural network for approximating a product function of p piecewise multivariate linear functions. The existence of such a deep neural network with O(n + p) layers and O(dn + dnp + dp) neurons has been proven, where d is the number of variables of the multivariate function being approximated, 𝜖 is the approximation error, and n = O(p + log2(pd/𝜖)). For the second component, this study proposes a general propagational reinforcement-learning-based learning search method that improves the estimate h(.) according to the newly observed distance information about the goals, propagates the improvement bidirectionally in the search tree, and consequently obtains a sequence of more accurate V-values for a sequence of states. Experiments on the maze problems show that our method increases the convergence rate of reinforcement learning by a factor of 2.06 and reduces the number of learning episodes to 1/4 that of other nonpropagating methods.

With the continuous deepening of scientific development and technological innovation, many algorithms have been derived from the research on robot path planning, including the current research on ant colony algorithm. It is often accompanied by a series of problems such as slow convergence speed, seeking optimal solutions according to local conditions, insufficient optimization ability, and inability to complement the diversity of ant colony species and convergence speed. In view of the current shortcomings, first, the initial pheromone distribution is uneven, which can avoid the useless search behavior trajectory of the ants during the movement process, thereby speeding up the solution speed and saving time; secondly, based on the advantages of the A* algorithm, the introduction into its algorithm The heuristic search function further improves the heuristic function of the traditional ant colony algorithm and maximizes the speed of the search. Considering the influence factors of energy consumption factors on the experiment in actual situations, a new heuristic function is formed by combining with traditional factors, a simulation environment is built, and the proposed method is verified and compared through a set of examples. The results show that the improved ant colony algorithm is in the optimal path. It has better performance in length and number of iterations, and has practical engineering significance for robot path planning.KeywordsAnt colony algorithmA* algorithmPath planningMATLAB simulation

Constructive Polynomial Approximation in Sobolev Spaces

The performance and efficiency of A* search algorithm heavily depends on the quality of the heuristic function. Therefore, designing an optimal heuristic function becomes the primary goal of developing a search algorithm for specific domains in artificial intelligence. However, it is difficult to design a well-constructed heuristic function without careful consideration and trial-and-error, especially for complex pathfinding problems. The complexity of a heuristic function increases and becomes unmanageable to design when an increasing number of parameters are involved. Existing approaches often avoid complex heuristic function design: they either trade-off the accuracy for faster computation or taking advantage of the parallelism for better performance. The objective of this paper is to reduce the difficulty of complex heuristic function design for A* search algorithm. We aim to design an algorithm that can be automatically optimized to achieve rapid search with high accuracy and low computational cost. In this paper, we present a novel design and optimization method for a Multi-Weighted-Heuristics function (MWH) named Evolutionary Heuristic A* search (EHA*) to: (1) minimize the effort on heuristic function design via Genetic Algorithm (GA), (2) optimize the performance of A* search and its variants including but not limited to WA* and MHA*, and (3) guarantee the completeness and optimality. EHA* algorithm enables high performance searches and significantly simplifies the processing of heuristic design. We apply EHA* to multiple grid-based pathfinding benchmarks to evaluate the performance. Our experiment result shows that EHA* (1) is capable of choosing an accurate heuristic function that provides an optimal solution, (2) can identify and eliminate inefficient heuristics, (3) is able to automatically design multi-heuristics function, and (4) minimizes both the time and space complexity.

Path planning for a mobile robot is a vital task for navigating in a complex environment. However, the path planning problem is challenging because of its non-deterministic polynomial-time (NP) character. In this research, the authors have proposed the Multi-objective Adaptive Ant Colony Optimization for path planning of mobile robot in a static object environment on the grid. The heuristic information function in the conventional ant colony optimization is modified according to the A* algorithm, which helps in mitigating the slow convergence of the traditional algorithm. It is because the ants will choose the nodes which are closer to the goal position. Additionally, an objective function is formulated as a multiple objective problem by incorporating (a) Path length, (b) Safety factor, and (c) Energy consumption. Simulation results show that the proposed modification helps to achieve the optimal path in a quicker time i.e., 1.3 times faster than the traditional counterpart.