13,099 research outputs found
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
MM Algorithms for Geometric and Signomial Programming
This paper derives new algorithms for signomial programming, a generalization
of geometric programming. The algorithms are based on a generic principle for
optimization called the MM algorithm. In this setting, one can apply the
geometric-arithmetic mean inequality and a supporting hyperplane inequality to
create a surrogate function with parameters separated. Thus, unconstrained
signomial programming reduces to a sequence of one-dimensional minimization
problems. Simple examples demonstrate that the MM algorithm derived can
converge to a boundary point or to one point of a continuum of minimum points.
Conditions under which the minimum point is unique or occurs in the interior of
parameter space are proved for geometric programming. Convergence to an
interior point occurs at a linear rate. Finally, the MM framework easily
accommodates equality and inequality constraints of signomial type. For the
most important special case, constrained quadratic programming, the MM
algorithm involves very simple updates.Comment: 16 pages, 1 figur
A hierarchical time-splitting approach for solving finite-time optimal control problems
We present a hierarchical computation approach for solving finite-time
optimal control problems using operator splitting methods. The first split is
performed over the time index and leads to as many subproblems as the length of
the prediction horizon. Each subproblem is solved in parallel and further split
into three by separating the objective from the equality and inequality
constraints respectively, such that an analytic solution can be achieved for
each subproblem. The proposed solution approach leads to a nested decomposition
scheme, which is highly parallelizable. We present a numerical comparison with
standard state-of-the-art solvers, and provide analytic solutions to several
elements of the algorithm, which enhances its applicability in fast large-scale
applications
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