41 - 48 of 48 results for: optimization methods

Optimization theory and modeling. The role of prices, duality, optimality conditions, and algorithms in finding and recognizing solutions. Perspectives: problem formulation, analytical theory, computational methods, and recent applications in engineering, finance, and economics. Theories: finite dimensional derivatives, convexity, optimality, duality, and sensitivity. Methods: simplex and interior-point, gradient, Newton, and barrier. Prerequisite: CME 100 or MATH 51 or equivalent.

Introduction of core algorithmic techniques and proof strategies that underlie the best known provable guarantees for minimizing high dimensional convex functions. Focus on broad canonical optimization problems and survey results for efficiently solving them, ultimately providing the theoretical foundation for further study in optimization. In particular, focus will be on first-order methods for both smooth and non-smooth convex function minimization as well as methods for structured convex function minimization, discussing algorithms such as gradient descent, accelerated gradient descent, mirror descent, Newton's method, interior point methods, and more. Prerequisite: multivariable calculus and linear algebra.

US health care spending is approximately 18% of GDP, growing rapidly, and driven in large part by prices and waste rather than quality and access. New approaches for improving health care delivery are urgently needed. This class focuses on the use of analytical tools to support efficient health care delivery. Topics include case studies on capacity planning, resource allocation, and scheduling. Methods include queueing, optimization, and simulation. Prerequisites: basic knowledge of Excel, probability, and optimization. For students in the Schools of Medicine, Business, and Law the course includes a variant of the curriculum with less emphasis on the technical material.

Formulation of standard linear programming models. Theory of polyhedral convex sets, linear inequalities, alternative theorems, and duality. Variants of the simplex method and the state of art interior-point algorithms. Sensitivity analyses, economic interpretations, and primal-dual methods. Relaxations of harder optimization problems and recent convex conic linear programs. Applications include game equilibrium facility location. Prerequisite: MATH 113 or consent of instructor.

Fundamental theory for solving continuous optimization problems with provable efficiency guarantees. Coverage of both canonical optimization methods and techniques, e.g. gradient descent, mirror descent, stochastic methods, acceleration, higher-order methods, etc. and canonical optimization problems, critical point computation for non-convex functions, smooth-convex function minimization, regression, linear programming, etc. Focus on provable rates for solving broad classes of prevalent problems including both classic problems and those motivated by large-scale computational concerns. Discussion of computational ramifications, fundamental information-theoretic limits, and problem structure. Prerequisite: linear algebra, multivariable calculus, probability, and proofs.

Emphasis is on the theoretical foundations of simulation methodology. Generation of uniform and non-uniform random variables. Discrete-event simulation and generalized semi-Markov processes. Output analysis (autoregressive, regenerative, spectral, and stationary times series methods). Variance reduction techniques (antithetic variables, common random numbers, control variables, discrete-time, conversion, importance sampling). Stochastic optimization (likelihood ratio method, perturbation analysis, stochastic approximation). Simulation in a parallel environment. Prerequisite: MS&E 221 or equivalent.

How do we use mathematical thinking to design better machine learning methods? This course focuses on developing mathematical tools for answering this question. This course will cover fundamental concepts and principled algorithms in machine learning, particularly those that are related to modern large-scale non-linear models. The topics include concentration inequalities, generalization bounds via uniform convergence, non-convex optimization, implicit regularization effect in deep learning, and unsupervised learning and domain adaptations. Prerequisites: linear algebra ( MATH 51 or CS 205), probability theory ( STATS 116, MATH 151 or CS 109), and machine learning ( CS 229, STATS 229, or STATS 315A).