On quasi $$\epsilon $$-solution for robust convex optimization problems

This paper devotes to the quasi \(\epsilon \)-solution (one sort of approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish approximate optimality theorem and approximate duality theorems in term of Wolfe type on quasi \(\epsilon \)-solution for the robust convex optimization problem. Moreover, some examples are given to illustrate the obtained results.     

A linear-time algorithm for the trust region subproblem based on hidden convexity

We present a linear-time approximation scheme for solving the trust region subproblem (TRS). It employs Nesterov’s accelerated gradient descent algorithm to solve a convex programming reformulation of (TRS). The total time complexity is less than that of the recent linear-time algorithm. The algorithm is further extended to the two-sided trust region subproblem.     

An exact algorithm to extend lifetime through roles allocation in sensor networks with connectivity constraints

We face the problem of scheduling optimally the activities in a wireless sensor network in order to ensure that, in each instant of time, the activated sensors can monitor all points of interest (targets) and route the collected information to a processing facility. Each sensor is allocated to a role, depending on whether it is actually used to monitor the targets, to forward information or kept idle, leading to different battery consumption ratios. We propose a column generation algorithm that embeds a highly efficient genetic metaheuristic for the subproblem. Moreover, to optimally solve the subproblem, we introduce a new formulation with fewer integer variables than a previous one proposed in the literature. Finally, we propose a stopping criterion to interrupt the optimal resolution of the subproblem as soon as a favorable solution is found. The results of our computational tests show that our algorithm consistently outperforms previous approaches in the literature, and also improves the best results known to date on some benchmark instances.     

On metric subregularity for convex constraint systems by primal equivalent conditions

In this paper, we mainly study metric subregularity for a convex constraint system defined by a convex set-valued mapping and a convex constraint subset. The main work is to provide several primal equivalent conditions for metric subregularity by contingent cone and graphical derivative. Further it is proved that these primal equivalent conditions can characterize strong basic constraint qualification of convex constraint system given by Zheng and Ng (SIAM J Optim 18:437–460, 2007).     

On optimal partitions,individual values and cooperative games: Does a wiser agent always produce a higher value?

We consider an optimal partition of resources (e.g. consumers) between several agents, given utility functions (“wisdoms”) for the agents and their capacities. This problem is a variant of optimal transport (Monge–Kantorovich) between two measure spaces where one of the measures is discrete (capacities) and the costs of transport are the wisdoms of the agents. We concentrate on the individual value for each agent under optimal partition and show that, counter-intuitively, this value may decrease if the agent’s wisdom is increased. Sufficient and necessary conditions for the monotonicity with respect to the wisdom functions of the individual values will be given, independently of the other agents. The sharpness of these conditions is also discussed. Motivated by the above we define a cooperative game based on optimal partition and investigate conditions for stability of the grand coalition.     

Hedging with temporary price impact

We consider the problem of hedging a European contingent claim in a Bachelier model with temporary price impact as proposed by Almgren and Chriss (J Risk 3:5–39, 2001). Following the approach of Rogers and Singh (Math Financ 20:597–615, 2010) and Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b) from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets as, e.g., Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b), Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), Rogers and Singh (Math Financ 20:597–615, 2010), Almgren and Li (Option hedging with smooth market impact. Preprint, 2015), Moreau et al. (Math Financ. doi: 10.1111/mafi.12098, 2015), Kallsen and Muhle-Karbe (High-resilience limits of block-shaped order books. Preprint, 2014), Guasoni and Weber (Mathematical Financ. doi: 10.1111/mafi.12099, 2015a; Nonlinear price impact and portfolio choice. Preprint, 2015b), where the frictionless hedging strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods.     

Arbitrage without borrowing or short selling?

We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwise Riemann–Stieltjes integral. Our result exemplifies the potential intricacies of formulating economically meaningful self-financing conditions in continuous time, when one leaves the conventional arbitrage-free framework.     

Unsupervised and semi-supervised extreme learning machine with wavelet kernel for high dimensional data

Extreme learning machine (ELM) not only is an effective classifier in supervised learning, but also can be applied on unsupervised learning and semi-supervised learning. The model structure of unsupervised extreme learning machine (US-ELM) and semi-supervised extreme learning machine (SS-ELM) are same as ELM, the difference between them is the cost function. We introduce kernel function to US-ELM and propose unsupervised extreme learning machine with kernel (US-KELM). And SS-KELM has been proposed. Wavelet analysis has the characteristics of multivariate interpolation and sparse change, and Wavelet kernel functions have been widely used in support vector machine. Therefore, to realize a combination of the wavelet kernel function, US-ELM, and SS-ELM, unsupervised extreme learning machine with wavelet kernel function (US-WKELM) and semi-supervised extreme learning machine with wavelet kernel function (SS-WKELM) are proposed in this paper. The experimental results show the feasibility and validity of US-WKELM and SS-WKELM in clustering and classification.     

Value sharing problems for differential and difference polynomials of meromorphic functions in a non-Archimedean field

In this paper we discuss the uniqueness problem for differential and difference polynomials of the form (f ^nm (z)f ^nd (qz + c))^(k) for meromorphic functions in a non-Archimedean field.