A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming.[1] It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices.[citation The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory; though the basic concepts of dynamic programming are prefigured in John von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior and Abraham Wald's sequential analysis.[citation In continuous-time optimization problems, the analogous equation is a partial differential equation that is called the Hamilton–Jacobi–Bellman equation.[4][5] In discrete time any multi-stage optimization problem can be solved by analyzing the appropriate Bellman equation.

Many of the commodities that affect our everyday lives are processed by smelters and fumer-furnace operations. These include products made from copper, nickel, and zinc, such as electrical wiring, kitchen appliances, and some types of batteries. Smelters are also used to produce precious metals, such as gold or silver, and animal feed, and to provide heat for dryers and roasters, which must be operated in concert for efficient production. Despite the importance of smelters, operations are becoming increasingly challenging. Most smelters globally have been in operation for at least 20 years, resulting in higher maintenance requirements. Second, feed quality is declining.

Robots performing tasks in warehouses provide the first example of wide-spread adoption of autonomous vehicles in transportation and logistics. The efficiency of these operations, which can vary widely in practice, are a key factor in the success of supply chains. In this work we consider the problem of coordinating a fleet of robots performing picking operations in a warehouse so as to maximize the net profit achieved within a time period while respecting problem- and robot-specific constraints. We formulate the problem as a weighted set packing problem where the elements in consideration are items on the warehouse floor that can be picked up and delivered within specified time windows. We enforce the constraint that robots must not collide, that each item is picked up and delivered by at most one robot, and that the number of robots active at any time does not exceed the total number available. Since the set of routes is exponential in the size of the input, we attack optimization of the resulting integer linear program using column generation, where pricing amounts to solving an elementary resource-constrained shortest-path problem. We propose an efficient optimization scheme that avoids consideration of every increment within the time windows. We also propose a heuristic pricing algorithm that can efficiently solve the pricing subproblem. While this itself is an important problem, the insights gained from solving these problems effectively can lead to new advances in other time-widow constrained vehicle routing problems.

Bayesian experimental design (BED) is to answer the question that how to choose designs that maximize the information gathering. For implicit models, where the likelihood is intractable but sampling is possible, conventional BED methods have difficulties in efficiently estimating the posterior distribution and maximizing the mutual information (MI) between data and parameters. Recent work proposed the use of gradient ascent to maximize a lower bound on MI to deal with these issues. However, the approach requires a sampling path to compute the pathwise gradient of the MI lower bound with respect to the design variables, and such a pathwise gradient is usually inaccessible for implicit models. In this paper, we propose a novel approach that leverages recent advances in stochastic approximate gradient ascent incorporated with a smoothed variational MI estimator for efficient and robust BED. Without the necessity of pathwise gradients, our approach allows the design process to be achieved through a unified procedure with an approximate gradient for implicit models. Several experiments show that our approach outperforms baseline methods, and significantly improves the scalability of BED in high-dimensional problems.

Optimizing the hyperparameters and architecture of a neural network is a long yet necessary phase in the development of any new application. This consuming process can benefit from the elaboration of strategies designed to quickly discard low quality configurations and focus on more promising candidates. This work aims at enhancing HyperNOMAD, a library that adapts a direct search derivative-free optimization algorithm to tune both the architecture and the training of a neural network simultaneously, by targeting two keys steps of its execution and exploiting cheap approximations in the form of static surrogates to trigger the early stopping of the evaluation of a configuration and the ranking of pools of candidates. These additions to HyperNOMAD are shown to improve on its resources consumption without harming the quality of the proposed solutions.

We review our recent work in the area of autonomous materials research, highlighting the coupling of machine learning methods and models and more problem-aware modeling. We review the general Bayesian framework for closed-loop design employed by many autonomous materials platforms. We then provide examples of our work on such platforms. We finally review our approaches to extend current statistical and ML models to better reflect problem-specific structure including the use of physics-based models and incorporation of operational considerations into the decision-making procedure.

Abstract--Machine learning (ML) models are constructed by expert ML practitioners using various coding languages, in which they tune and select models hyperparameters and learning algorithms for a given problem domain. They also carefully design an objective function or loss function (often with multiple objectives) that captures the desired output for a given ML task such as classification, regression, etc. In multi-objective optimization, conflicting objectives and constraints is a major area of concern. In such problems, several competing objectives are seen for which no single optimal solution is found that satisfies all desired objectives simultaneously. In the past VA systems have allowed users to interactively construct objective functions for a classifier. In this paper, we extend this line of work by prototyping a technique to visualize multi-objective objective functions either defined in a Jupyter notebook or defined using an interactive visual interface to help users to: (1) perceive and interpret complex mathematical terms in it and (2) detect and resolve conflicting objectives. Visualization of the objective function enlightens potentially conflicting objectives that obstructs selecting correct solution(s) for the desired ML task or goal. We also present an enumeration of potential conflicts in objective specification in multi-objective objective functions for classifier selection. Furthermore, we demonstrate our approach in a VA system that helps users in specifying meaningful objective functions to a classifier by detecting and resolving conflicting objectives and constraints. Through a within-subject quantitative and qualitative user study, we present results showing that our technique helps users interactively specify meaningful objective functions by resolving potential conflicts for a classification task. In the past, researchers in visual analytics (VA) have investigated making ML model construction interactive, which means developing visual interfaces that allow users to construct ML models by interacting with graphical widgets or data marks [1], [2]. For example, the system XClusim helps biologists to interactively cluster a specified dataset [3], Hypermoval [4] and BEAMES [5] allows interactive construction of regression models, Axissketcher allows dimension reduction using simple drag-drop interactions [6]. Workflow adopted in the system CACTUS. Recently, Das et al. have demonstrated that may result into incorrectly predicting many relevant data a VA system, QUESTO [7] that facilitated interactive creation of instances, though improving the generalizability of the model. Here objective functions to solve a classification task utilising an Auto-the objective to train a model with high accuracy on a set of ML system.

In this paper, we present new image segmentation methods based on hidden Markov random fields (HMRFs) and cuckoo search (CS) variants. HMRFs model the segmentation problem as a minimization of an energy function. CS algorithm is one of the recent powerful optimization techniques. Therefore, five variants of the CS algorithm are used to compute a solution. Through tests, we conduct a study to choose the CS variant with parameters that give good results (execution time and quality of segmentation). CS variants are evaluated and compared with non-destructive testing (NDT) images using a misclassification error (ME) criterion.

Basin hopping is a global optimization algorithm. It was developed to solve problems in chemical physics, although it is an effective algorithm suited for nonlinear objective functions with multiple optima. In this tutorial, you will discover the basin hopping global optimization algorithm. Basin Hopping Optimization in Python Photo by Pedro Szekely, some rights reserved. Basin Hopping is a global optimization algorithm developed for use in the field of chemical physics.

Abstract: The fields of machine learning and statistics have invested great efforts into designing algorithms, models, and approaches that better predict future observations. Larger and richer data have also been shown to improve predictive power. This is especially true in the world of human behavioral big data, as is evident from recent advances in behavioral prediction technology. Large internet platforms that collect behavioral big data predict user behavior for their internal commercial purposes as well as for third parties, such as advertisers, insurers, security forces, and political consulting firms, who utilize the predictions for user-level personalization, targeting, and other decision-making. While machine learning algorithmic and data efforts are directed at improving predicted values, the internet platforms can minimize prediction error by «pushing» users' actions towards their predicted values using behavior modification techniques.