Hedge algorithm and Dual Averaging schemesBaes, Michel; Bürgisser, Michael
doi: 10.1007/s00186-012-0418-1pmid: N/A
We show that the Hedge algorithm, a method that is widely used in Machine Learning, can be interpreted as a particular instance of Dual Averaging schemes, which have recently been introduced by Nesterov for regret minimization. Based on this interpretation, we establish three alternative methods of the Hedge algorithm: one in the form of the original method, but with optimal parameters, one that requires less a priori information, and one that is better adapted to the context of the Hedge algorithm. All our modified methods have convergence results that are better or at least as good as the performance guarantees of the vanilla method. In numerical experiments, our methods significantly outperform the original scheme.
Certification aspects of the fast gradient method for solving the dual of parametric convex programsRichter, Stefan; Jones, Colin; Morari, Manfred
doi: 10.1007/s00186-012-0420-7pmid: N/A
This paper examines the computational complexity certification of the fast gradient method for the solution of the dual of a parametric convex program. To this end, a lower iteration bound is derived such that for all parameters from a compact set a solution with a specified level of suboptimality will be obtained. For its practical importance, the derivation of the smallest lower iteration bound is considered. In order to determine it, we investigate both the computation of the worst case minimal Euclidean distance between an initial iterate and a Lagrange multiplier and the issue of finding the largest step size for the fast gradient method. In addition, we argue that optimal preconditioning of the dual problem cannot be proven to decrease the smallest lower iteration bound. The findings of this paper are of importance in embedded optimization, for instance, in model predictive control.
On the approximability of adjustable robust convex optimization under uncertaintyBertsimas, Dimitris; Goyal, Vineet
doi: 10.1007/s00186-012-0405-6pmid: N/A
In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.
Portfolio-optimization models for small investorsBaumann, Philipp; Trautmann, Norbert
doi: 10.1007/s00186-012-0408-3pmid: N/A
Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.
An adaptive routing approach for personal rapid transitSchüpbach, Kaspar; Zenklusen, Rico
doi: 10.1007/s00186-012-0403-8pmid: N/A
Personal Rapid Transit (PRT) is a public transportation mode, in which small automated vehicles transport passengers on demand. Central control of the vehicles leads to interesting possibilities for optimized routings. The complexity of the involved routing problems together with the fact that routing algorithms for PRT essentially have to run in real-time often leads to the choice of fast greedy approaches. The most common routing approach is arguably a sequential one, where upcoming requests are greedily served in a quickest way without interfering with previously routed vehicles. The simplicity of this approach stems from the fact that a chosen route is never changed later. This is as well the main drawback of it, potentially leading to large detours. It is natural to ask how much one could gain by using a more adaptive routing strategy. This question is the main motivation of this article. In this paper, we first suggest a simple mathematical model for PRT, and then introduce a new adaptive routing algorithm that repeatedly uses solutions to an LP as a guide to route vehicles. Our routing approach incorporates new requests in the LP as soon as they appear, and reoptimizes the routing of all currently used vehicles, contrary to sequential routing. We provide preliminary computational results that give first evidence of the potential gains of an adaptive routing strategy, as used in our algorithm.
A polynomial time approximation algorithm for the two-commodity splittable flow problemEisenschmidt, Elke; Haus, Utz-Uwe
doi: 10.1007/s00186-012-0402-9pmid: N/A
We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity
$${i \in \{1, 2\}}$$
can be split into a bounded number k
i
of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k
i
and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k
1, k
2)-splittable flow without chunk size restrictions for fixed demand ratios.
Shape-preserving dynamic programmingCai, Yongyang; Judd, Kenneth
doi: 10.1007/s00186-012-0406-5pmid: N/A
Dynamic programming is the essential tool in dynamic economic analysis. Problems such as portfolio allocation for individuals and optimal growth of national economies are typical examples. Numerical methods typically approximate the value function and use value function iteration to compute the value function for the optimal policy. Polynomial approximations are natural choices for approximating value functions when we know that the true value function is smooth. However, numerical value function iteration with polynomial approximations is unstable because standard methods such as interpolation and least squares fitting do not preserve shape. We introduce shape-preserving approximation methods that stabilize value function iteration, and are generally faster than previous stable methods such as piecewise linear interpolation.
Tackling indeterminacy in overlapping generations modelsFeng, Zhigang
doi: 10.1007/s00186-012-0407-4pmid: N/A
Overlapping generations models may have a continuum of equilibria. Previous studies have been largely confined to the local analysis that linearizes the model around the steady state. However, what is true of the linearized system only applies for an unknown-sized open neighborhood of the steady state. In this paper I develop a method to diagnose the indeterminacy in overlapping generations models by computing the set of all equilibria. I also provide a procedure to simulate the economy with indeterminate equilibrium.