Abstract. 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 only accept 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 proven. 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.Key words. block coordinate descent, polynomial optimization problem, tensor form AMS subject classifications. 90C26, 90C30, 15A69, 49M271. Introduction. The optimization models whose objective and constraints are polynomial functions have recently attracted much research attention. This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful. For instance, even the simplest instances of polynomial optimization, such as maximizing a cubic polynomial over a sphere, is NP-hard (Nesterov [42]). However, the problem is so elementary that it can be even attempted in an undergraduate calculus class. For readers interested in polynomial optimization with simple constraints, we refer to De Klerk [12] for a survey on the computational complexity of optimizing various classes of polynomial functions over a simplex, hypercube or sphere. In particular, De Klerk et al.[13] designed a polynomial-time approximation scheme (PTAS) for minimizing polynomials of fixed degree over the simplex.So far, a few results have been obtained for approximation algorithms with guaranteed worst-case performance ratios for higher degree polynomial optimization problems. Luo and Zhang [39] derived a polynomial-time approximation algorithm to optimize a multivariate quartic polynomial over a region defined by quadratic inequalities.

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 models. 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.

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 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(n lg n), and more generally, for any constant r, r-exponential backoff has makespan Θ(n log lg r n). 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(n lg lg n/ lg lg lg 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 argument.In the adversarial-queuing model, we present the following stability and instability results for exponential backoff and loglogiterated 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/ lg n) and unstable for arrival rates of λ = Ω(lg lg n/ lg n); loglog-iterated backoff is stable for arrival rates of λ = O(1/(lg lg n lg n)) and unstable for arrival rates of λ = Ω(1/ lg n). 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.

Graph convolutional neural networks (GCNN) have become an increasingly active field of research. It models the spatial dependencies of nodes in a graph with a pre-defined Laplacian matrix based on node distances. However, in many application scenarios, spatial dependencies change over time, and the use of fixed Laplacian matrix cannot capture the change. To track the spatial dependencies among traffic data, we propose a dynamic spatio-temporal GCNN for accurate traffic forecasting. The core of our deep learning framework is the finding of the change of Laplacian matrix with a dynamic Laplacian matrix estimator. To enable timely learning with a low complexity, we creatively incorporate tensor decomposition into the deep learning framework, where real-time traffic data are decomposed into a global component that is stable and depends on long-term temporal-spatial traffic relationship and a local component that captures the traffic fluctuations. We propose a novel design to estimate the dynamic Laplacian matrix of the graph with above two components based on our theoretical derivation, and introduce our design basis. The forecasting performance is evaluated with two realtime traffic datasets. Experiment results demonstrate that our network can achieve up to 25% accuracy improvement.

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.

In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{xIf one of A k 's is indefinite while others and C are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m 2 ) when F is the real line R, and by O(m) when F is the complex plane C. This result is an extension of the recent work of Luo et al. [8]. For (2), we show that the same ratio is bounded from below by O(1/ log m) for both the real and complex case, whenever all but one of A k 's are positive semidefinite while C can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal et al. [2]. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.

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Random sampling is a simple but powerful method in statistics and the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ R n . To do so, one may select a simpler (even finite) subset S 0 ⊆ S, randomly take some samples over S 0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f , S and S 0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξ i (1 ≤ i ≤ n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob f (ξ 1 , ξ 2 , · · · , ξ n ) ≥ c 1 n − d 2 F 1 ≥ c 2 , where c 1 , c 2 > 0 are two universal constants. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.