Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.

In this paper, the problem of maximizing a black-box function $f:\mathcal{X} \to \mathbb{R}$ is studied in the Bayesian framework with a Gaussian Process (GP) prior. In particular, a new algorithm for this problem is proposed, and high probability bounds on its simple and cumulative regret are established. The query point selection rule in most existing methods involves an exhaustive search over an increasingly fine sequence of uniform discretizations of $\mathcal{X}$. The proposed algorithm, in contrast, adaptively refines $\mathcal{X}$ which leads to a lower computational complexity, particularly when $\mathcal{X}$ is a subset of a high dimensional Euclidean space. In addition to the computational gains, sufficient conditions are identified under which the regret bounds of the new algorithm improve upon the known results. Finally an extension of the algorithm to the case of contextual bandits is proposed, and high probability bounds on the contextual regret are presented.

We consider an extension of the set covering problem (SCP) introducing (i)~multicover and (ii)~generalized upper bound (GUB)~constraints. For the conventional SCP, the pricing method has been introduced to reduce the size of instances, and several efficient heuristic algorithms based on such reduction techniques have been developed to solve large-scale instances. However, GUB constraints often make the pricing method less effective, because they often prevent solutions from containing highly evaluated variables together. To overcome this problem, we develop heuristic algorithms to reduce the size of instances, in which new evaluation schemes of variables are introduced taking account of GUB constraints. We also develop an efficient implementation of a 2-flip neighborhood local search algorithm that reduces the number of candidates in the neighborhood without sacrificing the solution quality. In order to guide the search to visit a wide variety of good solutions, we also introduce a path relinking method that generates new solutions by combining two or more solutions obtained so far. According to computational comparison on benchmark instances, the proposed method succeeds in selecting a small number of promising variables properly and performs quite effectively even for large-scale instances having hard GUB constraints.

Here is a common problem: you have some machine learning algorithm you want to use but it has these damn hyperparameters. These are numbers like weight decay magnitude, Gaussian kernel width, and so forth. The algorithm doesn't set them, instead, it's up to you to determine their values. If you don't set these parameters to "good" values the algorithm doesn't work. Well, here is a list of everything I've seen people do, listed in order of most to least common: Guess and Check: Listen to your gut, pick numbers that feel good and see if they work. Keep doing this until you are tired of doing it. Grid Search: Ask your computer to try a bunch of values spread evenly over some range. Random Search: Ask your computer to try a bunch of values by picking them randomly. Bayesian Optimization: Use a tool like MATLAB's bayesopt to automatically pick the best parameters, then find out Bayesian Optimization has more hyperparameters than your machine learning algorithm, get frustrated, and go back to using guess and check or grid search.

I assume the small difference is because of the approximation differences in R. However the results are very close. One important point to remember about gradient descent is how we choose the learning rate. If it is too small, it will take forever to converge but if it is too large, we will never find a minima. From the plot above it easily understood that, in the initial stages, we were able to take big steps. As we get closer to the minima the descent starts slowing down.

The way work load is shared between these stages depends on the particular approach in use. The reader unfamiliar with Petri nets needs only know the following: They are made of places, transitions, and tokens. A place can be seen as a token holder. When it contains one or more tokens, it is said to be marked. Transitions allow tokens to circulate from place to place.

The late 1990s saw a huge upswing in the demand for supply-chain management software. The rapid commercial adoption of these tools brought into sharp relief the complexity of automating supply-chain planning and optimization.

Using some real-world examples I illustrate the important role of multiobjective optimization in decision making and its interface with preference handling. I explain what optimization in the presence of multiple objectives means and discuss some of the most common methods of solving multiobjective optimization problems using transformations to single-objective optimization problems. Finally, I address linear and combinatorial optimization problems with multiple objectives and summarize techniques for solving them. Throughout the article I refer to the real-world examples introduced at the beginning. There are infinitely many ways to invest money and infinitely many possible radiotherapy treatments, but the number of feasible crew schedules is finite, albeit astronomical in practice.

We describe an application of AI search and information visualization techniques to the problem of school redistricting, in which students are assigned to home schools within a county or school district. This is a multicriteria optimization problem in which competing objectives, such as school capacity, busing costs, and socioeconomic distribution, must be considered. Because of the complexity of the decision-making problem, tools are needed to help end users generate, evaluate, and compare alternative school assignment plans. A key goal of our research is to aid users in finding multiple qualitatively different redistricting plans that represent different tradeoffs in the decision space. We present heuristic search methods that can be used to find a set of qualitatively different plans, and give empirical results of these search methods on population data from the school district of Howard County, Maryland.

I highlight several of these applications, using a simple energy storage problem as a case application. Using this setting, I describe a modeling framework that is based on five fundamental dimensions and that is more natural than the standard canonical form widely used in the reinforcement learning community. The framework focuses on finding the best policy, where I identify four fundamental classes of policies consisting of policy function approximations (PFAs), cost function approximations (CFAs), policies based on value function approximations (VFAs), and look-ahead policies. There is the familiar array of decisions: discrete actions, continuous controls, and vector-valued (and possibly integer) decisions. The tools for these problems are drawn from computer science, engineering, applied math, and operations research.