4,639 research outputs found
Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics
The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session
A D.C. Programming Approach to the Sparse Generalized Eigenvalue Problem
In this paper, we consider the sparse eigenvalue problem wherein the goal is
to obtain a sparse solution to the generalized eigenvalue problem. We achieve
this by constraining the cardinality of the solution to the generalized
eigenvalue problem and obtain sparse principal component analysis (PCA), sparse
canonical correlation analysis (CCA) and sparse Fisher discriminant analysis
(FDA) as special cases. Unlike the -norm approximation to the
cardinality constraint, which previous methods have used in the context of
sparse PCA, we propose a tighter approximation that is related to the negative
log-likelihood of a Student's t-distribution. The problem is then framed as a
d.c. (difference of convex functions) program and is solved as a sequence of
convex programs by invoking the majorization-minimization method. The resulting
algorithm is proved to exhibit \emph{global convergence} behavior, i.e., for
any random initialization, the sequence (subsequence) of iterates generated by
the algorithm converges to a stationary point of the d.c. program. The
performance of the algorithm is empirically demonstrated on both sparse PCA
(finding few relevant genes that explain as much variance as possible in a
high-dimensional gene dataset) and sparse CCA (cross-language document
retrieval and vocabulary selection for music retrieval) applications.Comment: 40 page
New complexity results and algorithms for min-max-min robust combinatorial optimization
In this work we investigate the min-max-min robust optimization problem
applied to combinatorial problems with uncertain cost-vectors which are
contained in a convex uncertainty set. The idea of the approach is to calculate
a set of k feasible solutions which are worst-case optimal if in each possible
scenario the best of the k solutions would be implemented. It is known that the
min-max-min robust problem can be solved efficiently if k is at least the
dimension of the problem, while it is theoretically and computationally hard if
k is small. While both cases are well studied in the literature nothing is
known about the intermediate case, namely if k is smaller than but close to the
dimension of the problem. We approach this open question and show that for a
selection of combinatorial problems the min-max-min problem can be solved
exactly and approximately in polynomial time if some problem specific values
are fixed. Furthermore we approach a second open question and present the first
implementable algorithm with oracle-pseudopolynomial runtime for the case that
k is at least the dimension of the problem. The algorithm is based on a
projected subgradient method where the projection problem is solved by the
classical Frank-Wolfe algorithm. Additionally we derive a branch & bound method
to solve the min-max-min problem for arbitrary values of k and perform tests on
knapsack and shortest path instances. The experiments show that despite its
theoretical impact the projected subgradient method cannot compete with an
already existing method. On the other hand the performance of the branch &
bound method scales very well with the number of solutions. Thus we are able to
solve instances where k is above some small threshold very efficiently
Momentum-inspired Low-Rank Coordinate Descent for Diagonally Constrained SDPs
We present a novel, practical, and provable approach for solving diagonally
constrained semi-definite programming (SDP) problems at scale using accelerated
non-convex programming. Our algorithm non-trivially combines acceleration
motions from convex optimization with coordinate power iteration and matrix
factorization techniques. The algorithm is extremely simple to implement, and
adds only a single extra hyperparameter -- momentum. We prove that our method
admits local linear convergence in the neighborhood of the optimum and always
converges to a first-order critical point. Experimentally, we showcase the
merits of our method on three major application domains: MaxCut, MaxSAT, and
MIMO signal detection. In all cases, our methodology provides significant
speedups over non-convex and convex SDP solvers -- 5X faster than
state-of-the-art non-convex solvers, and 9 to 10^3 X faster than convex SDP
solvers -- with comparable or improved solution quality.Comment: 10 pages, 8 figures, preprint under revie
Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems
Development of robust dynamical systems and networks such as autonomous
aircraft systems capable of accomplishing complex missions faces challenges due
to the dynamically evolving uncertainties coming from model uncertainties,
necessity to operate in a hostile cluttered urban environment, and the
distributed and dynamic nature of the communication and computation resources.
Model-based robust design is difficult because of the complexity of the hybrid
dynamic models including continuous vehicle dynamics, the discrete models of
computations and communications, and the size of the problem. We will overview
recent advances in methodology and tools to model, analyze, and design robust
autonomous aerospace systems operating in uncertain environment, with stress on
efficient uncertainty quantification and robust design using the case studies
of the mission including model-based target tracking and search, and trajectory
planning in uncertain urban environment. To show that the methodology is
generally applicable to uncertain dynamical systems, we will also show examples
of application of the new methods to efficient uncertainty quantification of
energy usage in buildings, and stability assessment of interconnected power
networks
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