We develop a framework for rendering photographic images, taking into account display limitations, so as to optimize perceptual similarity between the rendered image and the original scene. We formulate this as a constrained optimization problem, in which we minimize a measure of perceptual dissimilarity, the Normalized Laplacian Pyramid Distance (NLPD), which mimics the early stage transformations of the human visual system. When rendering images acquired with higher dynamic range than that of the display, we find that the optimized solution boosts the contrast of low-contrast features without introducing significant artifacts, yielding results of comparable visual quality to current state-of-the art methods with no manual intervention or parameter settings. We also examine a variety of other display constraints, including limitations on minimum luminance (black point), mean luminance (as a proxy for energy consumption), and quantized luminance levels (halftoning). Finally, we show that the method may be used to enhance details and contrast of images degraded by optical scattering (e.g. fog).

Markov state models (MSMs) and Master equation models are popular approaches to approximate molecular kinetics, equilibria, metastable states, and reaction coordinates in terms of a state space discretization usually obtained by clustering. Recently, a powerful generalization of MSMs has been introduced, the variational approach (VA) of molecular kinetics and its special case the time-lagged independent component analysis (TICA), which allow us to approximate slow collective variables and molecular kinetics by linear combinations of smooth basis functions or order parameters. While it is known how to estimate MSMs from trajectories whose starting points are not sampled from an equilibrium ensemble, this has not yet been the case for TICA and the VA. Previous estimates from short trajectories, have been strongly biased and thus not variationally optimal. Here, we employ Koopman operator theory and ideas from dynamic mode decomposition (DMD) to extend the VA and TICA to non-equilibrium data. The main insight is that the VA and TICA provide a coefficient matrix that we call Koopman model, as it approximates the underlying dynamical (Koopman) operator in conjunction with the basis set used. This Koopman model can be used to compute a stationary vector to reweight the data to equilibrium. From such a Koopman-reweighted sample, equilibrium expectation values and variationally optimal reversible Koopman models can be constructed even with short simulations. The Koopman model can be used to propagate densities, and its eigenvalue decomposition provide estimates of relaxation timescales and slow collective variables for dimension reduction. Koopman models are generalizations of Markov state models, TICA and the linear VA and allow molecular kinetics to be described without a cluster discretization.

In many applications of black-box optimization, one can evaluate multiple points simultaneously, e.g. when evaluating the performances of several different neural network architectures in a parallel computing environment. In this paper, we develop a novel batch Bayesian optimization algorithm --- the parallel knowledge gradient method. By construction, this method provides the one-step Bayes optimal batch of points to sample. We provide an efficient strategy for computing this Bayes-optimal batch of points, and we demonstrate that the parallel knowledge gradient method finds global optima significantly faster than previous batch Bayesian optimization algorithms on both synthetic test functions and when tuning hyperparameters of practical machine learning algorithms, especially when function evaluations are noisy.

We propose a generic framework based on a new stochastic variance-reduced gradient descent algorithm for accelerating nonconvex low-rank matrix recovery. Starting from an appropriate initial estimator, our proposed algorithm performs projected gradient descent based on a novel semi-stochastic gradient specifically designed for low-rank matrix recovery. Based upon the mild restricted strong convexity and smoothness conditions, we derive a projected notion of the restricted Lipschitz continuous gradient property, and prove that our algorithm enjoys linear convergence rate to the unknown low-rank matrix with an improved computational complexity. Moreover, our algorithm can be employed to both noiseless and noisy observations, where the optimal sample complexity and the minimax optimal statistical rate can be attained respectively. We further illustrate the superiority of our generic framework through several specific examples, both theoretically and experimentally.

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a Mixed Integer Linear Optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed in [8] and the continuous nonconvex quadratic optimization framework of [33]. We propose new discrete first-order methods, which when paired with state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular $\ell_{1}$-based approaches. We illustrate that our approach can handle problem instances with p = 10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling.

On Time Systems, Inc. (OTS) brings state of the art search-based optimization technology to industrial problems. In recent years, search-based techniques have surpassed the effectiveness of the long-dominant operations research techniques for solving a large variety of optimization problems, and OTS is at the leading edge.

Optimization problems are everywhere in engineering: Balancing design tradeoffs is an optimization problem, as are scheduling and logistical planning. The theory -- and sometimes the implementation -- of control systems relies heavily on optimization, and so does machine learning, which has been the basis of most recent advances in artificial intelligence. This week, at the IEEE Symposium on Foundations of Computer Science, a trio of present and past MIT graduate students won a best-student-paper award for a new "cutting-plane" algorithm, a general-purpose algorithm for solving optimization problems. The algorithm improves on the running time of its most efficient predecessor, and the researchers offer some reason to think that they may have reached the theoretical limit. But they also present a new method for applying their general algorithm to specific problems, which yields huge efficiency gains -- several orders of magnitude.

The development of problem solvers for configuration tasks is one of the most successful and mature application areas of artificial intelligence. The provision of tailored products, services, and systems requires efficient engineering and design processes where configurators play a crucial role. Because one of the core competencies of Siemens is to provide such highly engineered and customized systems, ranging from solutions for medium-sized and small businesses up to huge industrial plants, the efficient implementation and maintenance of configurators are important goals for the success of many departments. For more than 25 years the application of constraint-based methods has proven to be a key technology in order to realize configurators at Siemens. This article summarizes the main aspects and insights we have gained looking back over this period. In particular, we highlight the main technology factors regarding knowledge representation, reasoning, and integration which were important for our achievement. Finally we describe selected key application areas where the business success vitally depends on the high productivity of configuration processes.

The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range constraints. Split feasibility generalizes important inverse problems including convex feasibility, linear complementarity, and regression with constraint sets. When a feasible point does not exist, solution methods that proceed by minimizing a proximity function can be used to obtain optimal approximate solutions to the problem. We present an extension of the proximity function approach that generalizes the linear split feasibility problem to allow for non-linear mappings. Our algorithm is based on the principle of majorization-minimization, is amenable to quasi-Newton acceleration, and comes complete with convergence guarantees under mild assumptions. Furthermore, we show that the Euclidean norm appearing in the proximity function of the non-linear split feasibility problem can be replaced by arbitrary Bregman divergences. We explore several examples illustrating the merits of non-linear formulations over the linear case, with a focus on optimization for intensity-modulated radiation therapy.

In this paper, we propose two {\bf accelerated stochastic subgradient} methods for stochastic non-strongly convex optimization problems by leveraging a generic local error bound condition. The novelty of the proposed methods lies at smartly leveraging the recent historical solution to tackle the variance in the stochastic subgradient. The key idea of both methods is to iteratively solve the original problem approximately in a local region around a recent historical solution with size of the local region gradually decreasing as the solution approaches the optimal set. The difference of the two methods lies at how to construct the local region. The first method uses an explicit ball constraint and the second method uses an implicit regularization approach. For both methods, we establish the improved iteration complexity in a high probability for achieving an $\epsilon$-optimal solution. Besides the improved order of iteration complexity with a high probability, the proposed algorithms also enjoy a logarithmic dependence on the distance of the initial solution to the optimal set. We also consider applications in machine learning and demonstrate that the proposed algorithms enjoy faster convergence than the traditional stochastic subgradient method. For example, when applied to the $\ell_1$ regularized polyhedral loss minimization (e.g., hinge loss, absolute loss), the proposed stochastic methods have a logarithmic iteration complexity.