SDP-based Algorithm Research Articles

Source Localization Using TDOA Measurements From Underwater Acoustic Sensor Networks

In this letter, we investigate the problem of TDOA localization with unknown signal propagation speed in underwater acoustic sensor networks (UASNs). Although the state-of-the-art semidefinite programming (SDP)-based algorithm has good accuracy, its computational complexity is huge. We develop a two-step algorithm to estimate the source position and signal propagation speed. It not only has better accuracy than the SDP-based algorithm but also has much less computational complexity. The first step of the proposed algorithm is a least squares (LS) estimator, and the second step of the proposed algorithm is a fixed point iteration (FPI) estimator. Numerical simulations validate that the proposed algorithm is superior to the SDP algorithm and reaches the Cramér-Rao lower bounds (CRLB).

On the Robust design for IoT-based Wireless Information and Power Transmission network

This work investigates the robust beamforming for a multi-antenna internet of things (IoT) system using wireless information and power transmission (WIPT), given that imperfect channel state information (CSI) assumption is accounted.In particular, we investigate the problem of maximizing the worst-case energy harvested, taking into account the quality of service (QoS) constraint of user rate. The proposed problem is naturally a nonconvex problem, which is hard to tackle directly. On one hand, we rely on a classical method, that is semidefinite programming problem (SDP), to handle this by transforming the original nonconvex optimization problem with infinite number of constraints to a relaxed convex one. On the other hand, we propose another algorithm using Symbiotic Organisms Search (SOS) approach that can effciently solve the formulated problem. In the end, numerical results are provided to verify the effectiveness of the SDP-based algorithm in comparison with that of the SOS-based algorithm.

Stochastic Optimization of Braking Energy Storage and Ventilation in a Subway Station

In the Paris subway system, stations represent about one-third of the overall energy consumption. Within stations, ventilation is among the top consuming devices; it is operated at maximum airflow all day long, for air quality reasons. In this paper, we present a concept of energy system that displays comparable air quality while consuming much less energy. The system comprises a battery that makes it possible to recover the trains braking energy, arriving under the form of erratic and strong peaks. We propose an energy management system that, at short time scale, controls energy flows and ventilation airflow. By using proper optimization algorithms, we manage to match supply with demand, while minimizing energy daily costs. For this purpose, we have designed algorithms that take into account the braking variability. They are based on the so-called stochastic dynamic programming (SDP) mathematical framework. We fairly compare SDP-based algorithms with the widespread model predictive control (MPC) ones. First, both SDP and MPC yield energy/money operating savings of the order of one-third, compared to the current management without battery. Second, depending on the specific design, we observe that SDP outperforms MPC by a few percent, with an easier online numerical implementation.

Quantum Compression Relative to a Set of Measurements

In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and B\'ezout's theorem. The latter approach allows lifting the results from the single copy level to the case of multiple copies and from completely positive to merely positive maps.

Identifying a Set of Key Members in Social Networks Using SDP-Based Stochastic Search and Integer Programming Algorithms

This paper presents two semi-definite programming (SDP) based methods to solve the Key Player Problem (KPP). The KPP is to identify a set of [Formula: see text] nodes (i.e., key players) from a social network of size [Formula: see text] such that the number of nodes connected to these [Formula: see text] nodes is maximized. The KPP has applications in social diffusion and products adoption as it helps maximizing information diffusion and impact. We first formulate the KPP as an integer program (IP) and then convert it into an SDP formulation, which can be solved efficiently and produce a set of high quality candidate solutions. We develop an IP-based algorithm and a stochastic search (greedy) algorithm to find the final solution for the KPP. We compare our algorithms with existing methods in small and large networks with different network structures, including random graph, scale-free network, and community-based scale-free network (CSN). Computational results show that our algorithms are more efficient in solving the KPP in all networks. In addition, we examine how the network structure influences the nodes coverage. It is found that CSNs allow the highest nodes coverage due to their community and scale-free structure.

A Sum-of-Squares and Semidefinite Programming Approach for Maximum Likelihood DOA Estimation.

Direction of arrival (DOA) estimation using a uniform linear array (ULA) is a classical problem in array signal processing. In this paper, we focus on DOA estimation based on the maximum likelihood (ML) criterion, transform the estimation problem into a novel formulation, named as sum-of-squares (SOS), and then solve it using semidefinite programming (SDP). We first derive the SOS and SDP method for DOA estimation in the scenario of a single source and then extend it under the framework of alternating projection for multiple DOA estimation. The simulations demonstrate that the SOS- and SDP-based algorithms can provide stable and accurate DOA estimation when the number of snapshots is small and the signal-to-noise ratio (SNR) is low. Moveover, it has a higher spatial resolution compared to existing methods based on the ML criterion.

Layout Decomposition for Triple Patterning Lithography

As minimum feature size and pitch spacing further scale down, triple patterning lithography is a likely 193 nm extension along the paradigm of double patterning lithography for 14-nm technology node. Layout decomposition, which divides input layout into several masks to minimize the conflict and stitch numbers, is a crucial design step for double/triple patterning lithography. In this paper, we present a systematic study on triple patterning layout decomposition problem, which is shown to be NP-hard. Because of the NP-hardness, the runtime required to exactly solve it increases dramatically with the problem size. We first propose a set of graph division techniques to reduce the problem size. Then, we develop integer linear programming (ILP) to solve it. For large layouts, even with the graph-division techniques, ILP may still suffer from serious runtime overhead. To achieve better trade-off between runtime and performance, we present a novel semidefinite programming (SDP)-based algorithm. Followed by a mapping process, we can translate the SDP solutions into the final decomposition solutions. Experimental results show that the graph division can reduce runtime dramatically. In addition, SDP-based algorithm can achieve great speed-up even compared with accelerated ILP, with very comparable results in terms of the stitch number and the conflict number.

A Special Class of Fractional QCQP and Its Applications on Cognitive Collaborative Beamforming

In this paper, we investigate a special class of the fractional quadratically constrained quadratic problem (QCQP) with more than two quadratic constraints. We propose to effectively solve this special fractional QCQP by one or two convex semidefinite programmings (SDPs). For the two SDPs, one is equivalent to the original fractional QCQP with rank-one relaxation from the Charnes-Cooper transformation and the other, exploiting the optimal value of the former SDP, always has rank-one solution, which is optimal to the former SDP. Theoretical analysis shows that our proposed non-iterative SDP-based algorithm achieves the global optimal solution to the special fractional QCQP. In specific scenarios, our proposed non-iterative SDP-based algorithm has lower computational complexity compared to the second-order cone programming (SOCP)-based and constrained concave convex procedure (CCCP)-based iterative algorithms. We apply the proposed non-iterative SDP-based algorithm on two collaborative beamforming problems in cognitive relay networks, specifically, one is the achievable rate region for two-way non-regenerative cognitive relay networks and the other is the achievable secrecy rate for one-way regenerative cognitive relay networks. Simulation results have shown that our proposed non-iterative SDP-based algorithm achieves the same performance as the SOCP-based iterative algorithm. Our proposed algorithm achieves the better performance than the CCCP-based iterative algorithm.

SDP-based algorithms for maximum independent set problems on hypergraphs

This paper deals with approximations of maximum independent sets in non-uniform hypergraphs of low degree. We obtain the first performance ratio that is sublinear in terms of the maximum or average degree of the hypergraph. We extend this to the weighted case and give a O(D̄loglogD̄/logD̄) bound, where D̄ is the average weighted degree in a hypergraph, matching the best bounds known for the special case of graphs. Our approach is to use an semi-definite technique to sparsify a given hypergraph and then apply combinatorial algorithms to find a large independent set in the resulting sparser instance.

Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász–Schrijver Hierarchy

Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for Max Cut and Sparsest Cut use semidefinite programming. Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is Vertex Cover. Probabilistically checkable proof (PCP)-based techniques of Dinur and Safra [Ann. of Math./ (2), 162 (2005), pp. 439-486] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. There is a widespread belief that semidefinite programming (SDP) techniques are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [Theory Comput., 2 (2006), pp. 19-51], our aim is to show that a large family of linear programming (LP)- and SDP-based algorithms fail to produce an approximation for Vertex Cover better than 2. Lovasz and Schrijver [SIAM J. Optim., 1 (1991), pp. 166-190] introduced the systems $LS$ and $LS_+$ for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, $LS_+$ captures the celebrated SDP-based algorithms for Max Cut and Sparsest Cut mentioned above. We rule out polynomial-time SDP-based $2-\Omega(1)$ approximations for Vertex Cover using $LS_+$. In particular, for every $\epsilon>0$ we prove an integrality gap of $2-\epsilon$ for Vertex Cover SDPs obtained by tightening the standard LP relaxation with $\Omega(\sqrt{\log n/\log\log n})$ rounds of $LS_+$. While tight integrality gaps were known for Vertex Cover in the weaker $LS$ system [G. Schoenebeck, L. Trevisan, and M. Tulsiani, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2007, pp. 302-310], previous results did not rule out a $2-\Omega(1)$ approximation after even two rounds of $LS_+$.

Universal Rigidity and Edge Sparsification for Sensor Network Localization

Owing to their high accuracy and ease of formulation, there has been great interest in applying convex optimization techniques, particularly that of semidefinite programming (SDP) relaxation, to tackle the sensor network localization problem in recent years. However, a drawback of such techniques is that the resulting convex program is often expensive to solve. In order to speed up computation, various edge sparsification heuristics have been proposed, whose aim is to reduce the number of edges in the input graph. Although these heuristics do reduce the size of the convex program and hence make it faster to solve, they are often ad hoc in nature and do not preserve the localization properties of the input. As such, one often has to face a tradeoff between solution accuracy and computational effort. In this paper, we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input. At the heart of our heuristic is a graph decomposition procedure, which allows us to identify certain sparse generically universally rigid subgraphs of the input graph. Our computational results show that the proposed approach can significantly reduce the computational and memory complexities of SDP-based algorithms for solving the sensor network localization problem. Moreover, it compares favorably with existing speedup approaches, both in terms of accuracy and solution time.

Moment inequalities for sums of random matrices and their applications in optimization

In this paper, we consider various moment inequalities for sums of random matrices—which are well-studied in the functional analysis and probability theory literature—and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization. First, we show that the validity of a recent conjecture of Nemirovski is actually a direct consequence of the so-called non-commutative Khintchine’s inequality in functional analysis. Using this result, we show that an SDP-based algorithm of Nemirovski, which is developed for solving a class of quadratic optimization problems with orthogonality constraints, has a logarithmic approximation guarantee. This improves upon the polynomial approximation guarantee established earlier by Nemirovski. Furthermore, we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities. Secondly, we consider a recent result of Delage and Ye on the so-called data-driven distributionally robust stochastic programming problem. One of the assumptions in the Delage–Ye result is that the underlying probability distribution has bounded support. However, using a suitable moment inequality, we show that the result in fact holds for a much larger class of probability distributions. Given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems, we expect that the moment inequalities discussed in this paper will find further applications in optimization.

A branch-and-cut SDP-based algorithm for minimum sum-of-squares clustering

Minimum sum-of-squares clustering (MSSC) consists in partitioning a given set of n points into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. Recently, Peng & Xia (2005) established the equivalence between 0-1 semidefinite programming (SDP) and MSSC. In this paper, we propose a branch-and-cut algorithm for the underlying 0-1 SDP model. The algorithm obtains exact solutions for fairly large data sets with computing times comparable with those of the best exact method found in the literature.

Solving NP-hard semirandom graph problems in polynomial expected time

The aim of this paper is to present an SDP-based algorithm for finding a sparse induced subgraph of order Θ ( n ) hidden in a semirandom graph of order n. As an application we obtain an algorithm that requires not more than O ( n ) random edges in order to k-color a semirandom k-colorable graph within polynomial expected time, thereby extending results of Feige and Kilian [J. Comput. System Sci. 63 (2001) 639–671] and of Subramanian [J. Algorithms 33 (1999) 112–123].

On approximation of max-vertex-cover

We consider the max-vertex-cover (MVC) problem, i.e., find k vertices from an undirected and edge-weighted graph G=( V, E), where | V|= n⩾ k, such that the total edge weight covered by the k vertices is maximized. There is a 3/4-approximation algorithm for MVC, based on a linear programming relaxation. We show that the guaranteed ratio can be improved by a simple greedy algorithm for k>(3/4) n, and a simple randomized algorithm for k>(1/2) n. Furthermore, we study a semidefinite programming (SDP) relaxation based approximation algorithms for MVC. We show that, for a range of k, our SDP-based algorithm achieves the best performance guarantee among the four types of algorithms mentioned in this paper.