We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Convex Analysisの二,三の進展について

In this article, we have provided general, comprehensive coverage of the SDR technique, from its practical deployments and scope of applicability to key theoretical results. We have also showcased several representative applications, namely MIMO detection, B? shimming in MRI, and sensor network localization. Another important application, namely downlink transmit beamforming, is described in [1]. Due to space limitations, we are unable to cover many other beautiful applications of the SDR technique, although we have done our best to illustrate the key intuitive ideas that resulted in those applications. We hope that this introductory article will serve as a good starting point for readers who would like to apply the SDR technique to their applications, and to locate specific references either in applications or theory.

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Semidefinite Relaxation of Quadratic Optimization Problems

The widespread use of multisensor technology and the emergence of big data sets have highlighted the limitations of standard flat-view matrix models and the necessity to move toward more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift toward models that are essentially polynomial, the uniqueness of which, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints which match data properties and extract more general latent components in the data than matrix-based methods.

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Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis

The block coordinate descent (BCD) method is widely used for minimizing a continuous function $f$ of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of $f$ which are either locally tight upper bounds of $f$ or strictly convex local approximations of $f$. The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable or nonconvex. Our results unify and extend the existing convergence results ...

https://works.bepress.com/mingyi_hong/1/download/

A unified convergence analysis of block successive minimization methods for nonsmooth optimization

In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we accept only a block update that achieves the maximum improvement, hence the name of our new search method: maximum block improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proved. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem; thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.

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Maximum Block Improvement and Polynomial Optimization

This paper analyzes the worst-case performance of randomized backoff on simple multiple-access channels Most previous analysis of backoff has assumed a statistical arrival modelFor batched arrivals, in which all n packets arrive at time 0, we show the following tight high-probability bounds Randomized binary exponential backoff has makespan Θ(nlgn), and more generally, for any constant r, r-exponential backoff has makespan Θ(nloglgrn) Quadratic backoff has makespan Θ((n/lg n)3/2), and more generally, for r>1, r-polynomial backoff has makespan Θ((n/lg n)1+1/r) Thus, for batched inputs, both exponential and polynomial backoff are highly sensitive to backoff constants We exhibit a monotone superpolynomial subexponential backoff algorithm, called loglog-iterated backoff, that achieves makespan Θ(nlglg n/lglglg n) We provide a matching lower bound showing that this strategy is optimal among all monotone backoff algorithms Of independent interest is that this lower bound was proved with a delay sequence argumentIn the adversarial-queuing model, we present the following stability and instability results for exponential backoff and loglog-iterated backoff Given a (λ,T)-stream, in which at most n=λT packets arrive in any interval of size T, exponential backoff is stable for arrival rates of λ=O(1/lgn) and unstable for arrival rates of λ=Ω(lglgn/lgn); loglog-iterated backoff is stable for arrival rates of λ=O(1/(lglgn\lgn)) and unstable for arrival rates of λ=Ω(1/lgn) Our instability results show that bursty input is close to being worst-case for exponential backoff and variants and that even small bursts can create instabilities in the channel

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Adversarial contention resolution for simple channels

In this paper we develop tight bounds on the expected values of several risk measures that are of interest to us. This work is motivated by the robust optimization models arising from portfolio selection problems. Indeed, the whole paper is centered around robust portfolio models and solutions. The basic setting is to find a portfolio that maximizes (respectively, minimizes) the expected utility (respectively, disutility) values in the midst of infinitely many possible ambiguous distributions of the investment returns fitting the given mean and variance estimations. First, we show that the single-stage portfolio selection problem within this framework, whenever the disutility function is in the form of lower partial moments (LPM), or conditional value-at-risk (CVaR), or value-at-risk (VaR), can be solved analytically. The results lead to the solutions for single-stage robust portfolio selection models. Furthermore, the results also lead to a multistage adjustable robust optimization (ARO) solution when the disutility function is the second-order LPM. Exploring beyond the confines of convex optimization, we also consider the so-called S-shaped value function, which plays a key role in the prospect theory of Kahneman and Tversky. The nonrobust version of the problem is shown to be NP-hard in general. However, we present an efficient procedure for solving the robust counterpart of the same portfolio selection problem. In this particular case, the consideration of the robustness actually helps to reduce the computational complexity. Finally, we consider the situation whereby we have some additional information about the chance that a quadratic function of the random distribution reaches a certain threshold. That information helps to further reduce the ambiguity in the robust model. We show that the robust optimization problem in that case can be solved by means of semidefinite programming (SDP), if no more than two additional chance inequalities are to be incorporated.

Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{ x^* C x \mid x^* A_k x \ge 1, k=0,1,\ldots,m, x\in\mathbb{F}^n\}$ and (2) $\max \{ x^* C x \mid x^* A_k x \le 1, k=0,1,\ldots,m, x\in\mathbb{F}^n\}$, where $\mathbb{F}$ is either the real field $\mathbb{R}$ or the complex field $\mathbb{C}$, and $A_k,C$ are symmetric matrices. For the minimization model (1), we prove that if the matrix $C$ and all but one of the $A_k$'s are positive semidefinite, then the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by $O(m^2)$ when $\mathbb{F}=\mathbb{R}$, and by $O(m)$ when $\mathbb{F}=\mathbb{C}$. Moreover, when two or more of the $A_k$'s are indefinite, this ratio can be arbitrarily large. For the maximization model (2), we show that if $C$ and at most one of the $A_k$'s are indefinite while other $A_k$'s are positive semidefinite, then the ratio between the optimal value of (2) and its SDP relaxation is bounded from below by $O(1/\log m)$ for both the real and the complex case. This result improves the bound based on the so-called approximate S-Lemma of Ben-Tal, Nemirovski, and Roos [SIAM J. Optim., 13 (2002), pp. 535-560]. When two or more of the $A_k$'s in (2) are indefinite, we derive a general bound in terms of the problem data and the SDP solution. For both optimization models, we present examples to show that the derived approximation bounds are essentially tight.

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Simai He

Papers

Maximum Block Improvement and Polynomial Optimization

Adversarial contention resolution for simple channels

Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization