Simai He

Adversarial contention resolution for simple channels

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 model.For <i>batched arrivals</i>, in which all <i>n</i> packets arrive at time 0, we show the following tight high-probability bounds. Randomized binary exponential backoff has makespan Θ(<i>n</i>lg<i>n</i>), and more generally, for any constant <i>r</i>, <i>r</i>-exponential backoff has makespan Θ(<i>n</i>log^lg<i>r</i> <i>n</i>). Quadratic backoff has makespan Θ((<i>n</i>/lg <i>n</i>)^3/2), and more generally, for <i>r</i>>1, <i>r</i>-polynomial backoff has makespan Θ((<i>n</i>/lg <i>n</i>)^1+1/<i>r</i>). Thus, for batched inputs, both exponential and polynomial backoff are highly sensitive to backoff constants. We exhibit a monotone superpolynomial subexponential backoff algorithm, called <i>loglog-iterated backoff</i>, that achieves makespan Θ(<i>n</i>lglg <i>n</i>/lglglg <i>n</i>). 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 argument.In the adversarial-queuing model, we present the following stability and instability results for exponential backoff and loglog-iterated backoff. Given a (λ,<i>T</i>)-stream, in which at most <i>n</i>=λ<i>T</i> packets arrive in any interval of size <i>T</i>, exponential backoff is stable for arrival rates of λ=<i>O</i>(1/lg<i>n</i>) and unstable for arrival rates of λ=Ω(lglg<i>n</i>/lg<i>n</i>); loglog-iterated backoff is stable for arrival rates of λ=<i>O</i>(1/(lglg<i>n</i>\lg<i>n</i>)) and unstable for arrival rates of λ=Ω(1/lg<i>n</i>). 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.

Maximum Block Improvement and Polynomial 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.

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

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.

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

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)

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

\min \{ x^* C x \mid x^* A_k x \ge 1, k=0,1,\ldots,m, x\in\mathbb{F}^n\}

The cost of cache-oblivious searching

and (2)

Approximation Methods for Polynomial Optimization: Models, Algorithms, and Applications

\max \{ x^* C x \mid x^* A_k x \le 1, k=0,1,\ldots,m, x\in\mathbb{F}^n\}

Sorting by Length-Weighted Reversals: Dealing with Signs and Circularity

, where

Improved bounds on sorting by length-weighted reversals

\mathbb{F}

When all risk-adjusted performance measures are the same: in praise of the Sharpe ratio

is either the real field