936 research outputs found
A Primal-Dual Algorithmic Framework for Constrained Convex Minimization
We present a primal-dual algorithmic framework to obtain approximate
solutions to a prototypical constrained convex optimization problem, and
rigorously characterize how common structural assumptions affect the numerical
efficiency. Our main analysis technique provides a fresh perspective on
Nesterov's excessive gap technique in a structured fashion and unifies it with
smoothing and primal-dual methods. For instance, through the choices of a dual
smoothing strategy and a center point, our framework subsumes decomposition
algorithms, augmented Lagrangian as well as the alternating direction
method-of-multipliers methods as its special cases, and provides optimal
convergence rates on the primal objective residual as well as the primal
feasibility gap of the iterates for all.Comment: This paper consists of 54 pages with 7 tables and 12 figure
Implementing a smooth exact penalty function for equality-constrained nonlinear optimization
We develop a general equality-constrained nonlinear optimization algorithm
based on a smooth penalty function proposed by Fletcher (1970). Although it was
historically considered to be computationally prohibitive in practice, we
demonstrate that the computational kernels required are no more expensive than
other widely accepted methods for nonlinear optimization. The main kernel
required to evaluate the penalty function and its derivatives is solving a
structured linear system. We show how to solve this system efficiently by
storing a single factorization each iteration when the matrices are available
explicitly. We further show how to adapt the penalty function to the class of
factorization-free algorithms by solving the linear system iteratively. The
penalty function therefore has promise when the linear system can be solved
efficiently, e.g., for PDE-constrained optimization problems where efficient
preconditioners exist. We discuss extensions including handling simple
constraints explicitly, regularizing the penalty function, and inexact
evaluation of the penalty function and its gradients. We demonstrate the merits
of the approach and its various features on some nonlinear programs from a
standard test set, and some PDE-constrained optimization problems
Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
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