11,834 research outputs found
Performance of a community detection algorithm based on semidefinite programming
The problem of detecting communities in a graph is maybe one the most studied inference problems, given its simplicity and widespread diffusion among several disciplines. A very common benchmark for this problem is the stochastic block model or planted partition problem, where a phase transition takes place in the detection of the planted partition by changing the signal-to-noise ratio. Optimal algorithms for the detection exist which are based on spectral methods, but we show these are extremely sensible to slight modification in the generative model. Recently Javanmard, Montanari and Ricci-Tersenghi [1] have used statistical physics arguments, and numerical simulations to show that finding communities in the stochastic block model via semidefinite programming is quasi optimal. Further, the resulting semidefinite relaxation can be solved efficiently, and is very robust with respect to changes in the generative model. In this paper we study in detail several practical aspects of this new algorithm based on semidefinite programming for the detection of the planted partition. The algorithm turns out to be very fast, allowing the solution of problems with O(105) variables in few second on a laptop computer
Robust Transceiver Design for MISO Interference Channel with Energy Harvesting
In this paper, we consider multiuser multiple-input single-output (MISO)
interference channel where the received signal is divided into two parts for
information decoding and energy harvesting (EH), respectively. The transmit
beamforming vectors and receive power splitting (PS) ratios are jointly
designed in order to minimize the total transmission power subject to both
signal-to-interference-plus-noise ratio (SINR) and EH constraints. Most joint
beamforming and power splitting (JBPS) designs assume that perfect channel
state information (CSI) is available; however CSI errors are inevitable in
practice. To overcome this limitation, we study the robust JBPS design problem
assuming a norm-bounded error (NBE) model for the CSI. Three different solution
approaches are proposed for the robust JBPS problem, each one leading to a
different computational algorithm. Firstly, an efficient semidefinite
relaxation (SDR)-based approach is presented to solve the highly non-convex
JBPS problem, where the latter can be formulated as a semidefinite programming
(SDP) problem. A rank-one recovery method is provided to recover a robust
feasible solution to the original problem. Secondly, based on second order cone
programming (SOCP) relaxation, we propose a low complexity approach with the
aid of a closed-form robust solution recovery method. Thirdly, a new iterative
method is also provided which can achieve near-optimal performance when the
SDR-based algorithm results in a higher-rank solution. We prove that this
iterative algorithm monotonically converges to a Karush-Kuhn-Tucker (KKT)
solution of the robust JBPS problem. Finally, simulation results are presented
to validate the robustness and efficiency of the proposed algorithms.Comment: 13 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1407.0474 by other author
A Faster Quantum Algorithm for Semidefinite Programming via Robust IPM Framework
This paper studies a fundamental problem in convex optimization, which is to
solve semidefinite programming (SDP) with high accuracy. This paper follows
from existing robust SDP-based interior point method analysis due to [Huang,
Jiang, Song, Tao and Zhang, FOCS 2022]. While, the previous work only provides
an efficient implementation in the classical setting. This work provides a
novel quantum implementation.
We give a quantum second-order algorithm with high-accuracy in both the
optimality and the feasibility of its output, and its running time depending on
on well-conditioned instances. Due to the limitation of
quantum itself or first-order method, all the existing quantum SDP solvers
either have polynomial error dependence or low-accuracy in the feasibility
A Mixed-Integer SDP Solution Approach to Distributionally Robust Unit Commitment with Second Order Moment Constraints
A power system unit commitment (UC) problem considering uncertainties of
renewable energy sources is investigated in this paper, through a
distributionally robust optimization approach. We assume that the first and
second order moments of stochastic parameters can be inferred from historical
data, and then employed to model the set of probability distributions. The
resulting problem is a two-stage distributionally robust unit commitment with
second order moment constraints, and we show that it can be recast as a
mixed-integer semidefinite programming (MI-SDP) with finite constraints. The
solution algorithm of the problem comprises solving a series of relaxed MI-SDPs
and a subroutine of feasibility checking and vertex generation. Based on the
verification of strong duality of the semidefinite programming (SDP) problems,
we propose a cutting plane algorithm for solving the MI-SDPs; we also introduce
a SDP relaxation for the feasibility checking problem, which is an intractable
biconvex optimization. Experimental results on a IEEE 6-bus system are
presented, showing that without any tunings of parameters, the real-time
operation cost of distributionally robust UC method outperforms those of
deterministic UC and two-stage robust UC methods in general, and our method
also enjoys higher reliability of dispatch operation
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
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