1,948 research outputs found
The Convergent Generalized Central Paths for Linearly Constrained Convex Programming
The convergence of central paths has been a focal point of research on interior point methods. Quite detailed analyses have been made for the linear case. However, when it comes to the convex case, even if the constraints remain linear, the problem is unsettled. In [Math. Program., 103 (2005), pp. 63–94], Gilbert, Gonzaga, and Karas presented some examples in convex optimization, where the central path fails to converge. In this paper, we aim at finding some continuous trajectories which can converge for all linearly constrained convex optimization problems under some mild assumptions. We design and analyze a class of continuous trajectories, which are the solutions of certain ordinary differential equation (ODE) systems for solving linearly constrained smooth convex programming. The solutions of these ODE systems are named generalized central paths. By only assuming the existence of a finite optimal solution, we are able to show that, starting from any interior feasible point, (i) all of the generalized central paths are convergent, and (ii) the limit point(s) are indeed the optimal solution(s) of the original optimization problem. Furthermore, we illustrate that for the key example of Gilbert, Gonzaga, and Karas, our generalized central paths converge to the optimal solutions
Optimal curing policy for epidemic spreading over a community network with heterogeneous population
The design of an efficient curing policy, able to stem an epidemic process at
an affordable cost, has to account for the structure of the population contact
network supporting the contagious process. Thus, we tackle the problem of
allocating recovery resources among the population, at the lowest cost possible
to prevent the epidemic from persisting indefinitely in the network.
Specifically, we analyze a susceptible-infected-susceptible epidemic process
spreading over a weighted graph, by means of a first-order mean-field
approximation. First, we describe the influence of the contact network on the
dynamics of the epidemics among a heterogeneous population, that is possibly
divided into communities. For the case of a community network, our
investigation relies on the graph-theoretical notion of equitable partition; we
show that the epidemic threshold, a key measure of the network robustness
against epidemic spreading, can be determined using a lower-dimensional
dynamical system. Exploiting the computation of the epidemic threshold, we
determine a cost-optimal curing policy by solving a convex minimization
problem, which possesses a reduced dimension in the case of a community
network. Lastly, we consider a two-level optimal curing problem, for which an
algorithm is designed with a polynomial time complexity in the network size.Comment: to be published on Journal of Complex Network
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
Analysis of the primal-dual central path for nonlinear semidefinite optimization without the nondegeneracy condition
We study properties of the central path underlying a nonlinear semidefinite
optimization problem, called NSDP for short. The latest radical work on this
topic was contributed by Yamashita and Yabe (2012): they proved that the
Jacobian of a certain equation-system derived from the Karush-Kuhn-Tucker (KKT)
conditions of the NSDP is nonsingular at a KKT point under the second-order
sufficient condition (SOSC), the strict complementarity condition (SC), and the
nondegeneracy condition (NC). This yields uniqueness and existence of the
central path through the implicit function theorem. In this paper, we consider
the following three assumptions on a KKT point: the strong SOSC, the SC, and
the Mangasarian-Fromovitz constraint qualification. Under the absence of the
NC, the Lagrange multiplier set is not necessarily a singleton and the
nonsingularity of the above-mentioned Jacobian is no longer valid. Nonetheless,
we establish that the central path exists uniquely, and moreover prove that the
dual component of the path converges to the so-called analytic center of the
Lagrange multiplier set. As another notable result, we clarify a region around
the central path where Newton's equations relevant to primal-dual interior
point methods are uniquely solvable
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