46 research outputs found
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
内点法・情報幾何・最適化モデリング
要旨あり最適化技術に基づく統計的推論研究詳
Recommended from our members
When Are Nonconvex Optimization Problems Not Scary?
Nonconvex optimization is NP-hard, even the goal is to compute a local minimizer. In applied disciplines, however, nonconvex problems abound, and simple algorithms, such as gradient descent and alternating direction, are often surprisingly effective. The ability of simple algorithms to find high-quality solutions for practical nonconvex problems remains largely mysterious.
This thesis focuses on a class of nonconvex optimization problems which CAN be solved to global optimality with polynomial-time algorithms. This class covers natural nonconvex formulations of central problems in signal processing, machine learning, and statistical estimation, such as sparse dictionary learning (DL), generalized phase retrieval (GPR), and orthogonal tensor decomposition. For each of the listed problems, the nonconvex formulation and optimization lead to novel and often improved computational guarantees.
This class of nonconvex problems has two distinctive features: (i) All local minimizer are also global. Thus obtaining any local minimizer solves the optimization problem; (ii) Around each saddle point or local maximizer, the function has a negative directional curvature. In other words, around these points, the Hessian matrices have negative eigenvalues. We call smooth functions with these two properties (qualitative) X functions, and derive concrete quantities and strategy to help verify the properties, particularly for functions with random inputs or parameters. As practical examples, we establish that certain natural nonconvex formulations for complete DL and GPR are X functions with concrete parameters.
Optimizing X functions amounts to finding any local minimizer. With generic initializations, typical iterative methods at best only guarantee to converge to a critical point that might be a saddle point or local maximizer. Interestingly, the X structure allows a number of iterative methods to escape from saddle points and local maximizers and efficiently find a local minimizer, without special initializations. We choose to describe and analyze the second-order trust-region method (TRM) that seems to yield the strongest computational guarantees. Intuitively, second-order methods can exploit Hessian to extract negative curvature directions around saddle points and local maximizers, and hence are able to successfully escape from the saddles and local maximizers of X functions. We state the TRM in a Riemannian optimization framework to cater to practical manifold-constrained problems. For DL and GPR, we show that under technical conditions, the TRM algorithm finds a global minimizer in a polynomial number of steps, from arbitrary initializations
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size