Iterative reweighted methods for $$\ell _1-\ell _p$$ minimization

In this paper, we focus on the \(\ell _1-\ell _p\) minimization problem with \(0<p<1\), which is challenging due to the \(\ell _p\) norm being non-Lipschizian. In theory, we derive computable lower bounds for nonzero entries of the generalized first-order stationary points of \(\ell _1-\ell _p\) minimization, and hence of its local minimizers. In algorithms, based on three locally Lipschitz continuous \(\epsilon \)-approximation to \(\ell _p\) norm, we design several iterative reweighted \(\ell _1\) and \(\ell _2\) methods to solve those approximation problems. Furthermore, we show that any accumulation point of the sequence generated by these methods is a generalized first-order stationary point of \(\ell _1-\ell _p\) minimization. This result, in particular, applies to the iterative reweighted \(\ell _1\) methods based on the new Lipschitz continuous \(\epsilon \)-approximation introduced by Lu (Math Program 147(1–2):277–307, 2014), provided that the approximation parameter \(\epsilon \) is below a threshold value. Numerical results are also reported to demonstrate the efficiency of the proposed methods.     

Finitely convergent cutting planes for concave minimization

In 1964 Tuy introduced a new type of cutting plane, the concavity cut, in the context of concave minimization. These cutting planes, which are also known as convexity cuts, intersection cuts and Tuy cuts, have found application in several algorithms, e.g., branch and bound algorithm, conical algorithm and cutting plane algorithm, and also in algorithms for other optimization problems, e.g., reverse convex programming, bilinear programming and Lipschitzian optimization. Up to now, however, it has not been possible to either prove or disprove the finite convergence of a pure cutting plane algorithm for concave minimization based solely on these cutting planes. In the present paper a modification of the concavity cut is proposed that yields deeper cutting planes and ensures the finite convergence of a pure cutting plane algorithm based on these cuts.     

An Exact Procedure for the Resource-Constrained Weighted Earliness–Tardiness Project Scheduling Problem

In this paper we study the resource-constrained project scheduling problem with weighted earliness–tardinesss penalty costs. Project activities are assumed to have a known deterministic due date, a unit earliness as well as a unit tardiness penalty cost and constant renewable resource requirements. The objective is to schedule the activities in order to minimize the total weighted earliness–tardinesss penalty cost of the project subject to the finish–start precedence constraints and the constant renewable resource availability constraints. With these features the problem becomes highly attractive in just-in-time environments.We introduce a depth-first branch-and-bound algorithm which makes use of extra precedence relations to resolve resource conflicts and relies on a fast recursive search algorithm for the unconstrained weighted earliness–tardinesss problem to compute lower bounds. The procedure has been coded in Visual C++, version 4.0 under Windows NT. Both the recursive search algorithm and the branch-and-bound procedure have been validated on a randomly generated problem set.     

Price competition and store competition: Store brands vs. national brand

This paper studies pricing strategies in a market channel composed of one national brand manufacturer and two retailers who, each, carry their own store brand and a national brand products. The model accounts for product competition between store brands and the national brand products, as well as for store competition between retailers.     

Pricing and advertisement in a manufacturer–retailer supply chain

We use a game theoretical approach to study pricing and advertisement decisions in a manufacturer–retailer supply chain when price discounts are offered by both the manufacturer and retailer. When the manufacturer is the leader of the game, we obtained Stackelberg equilibrium with manufacturer’s local allowance, national brand name investment, manufacturer’s preferred price discount, retailer’s price discount, and local advertising expense. For the special case of two-stage equilibrium when the manufacturer’s price discount is exogenous, we found that the retailer is willing to increase local advertising expense if the manufacturer increases local advertising allowance and provides deeper price discount, or if the manufacturer decreases its brand name investment. When both the manufacturer and retailer have power, Nash equilibrium in a competition game is obtained. The comparison between the Nash equilibrium and Stackelberg equilibrium shows that the manufacturer always prefers Stackelberg equilibrium, but there is no definitive conclusion for the retailer. The bargaining power can be used to determine the profit sharing between the manufacturer and the retailer. Once the profit sharing is determined, we suggest a simple contract to help the manufacturer and retailer obtain their desired profit sharing.     

Nonparametric Statistical Methods and the Pricing of Derivative Securities

In this review paper we summarise several nonparametric methods recently applied to the pricing of financial options. After a short introduction to martingale-based option pricing theory, we focus on two possible fields of application for nonparametric methods: the estimation of risk-neutral probabilities and the estimation of the dynamics of the underlying instruments in order to construct an internally consistent model.     

An admissible minimax estimator of a bounded scale-parameter in a subclass of the exponential family under scale-invariant squared-error loss

A subclass of the scale-parameter exponential family is considered and for the rth power of the scale parameter, which is lower bounded, an admissible minimax estimator under scale-invariant squared-error loss is presented. Also, an admissible minimax estimator of a lower-bounded parameter in the family of transformed chi-square distributions is given. These estimators are the pointwise limits of a sequence of Bayes estimators. Some examples are given.     

An Iterative Procedure for Evaluating Digraph Competitions

A competition which is based on the results of (partial) pairwise comparisons can be modelled by means of a directed graph. Given initial weights on the nodes in such digraph competitions, we view the measurement of the importance (i.e., the cardinal ranking) of the nodes as an allocation problem where we redistribute the initial weights on the basis of insights from cooperative game theory. After describing the resulting procedure of redistributing the initial weights, an iterative process is described that repeats this procedure: at each step the allocation obtained in the previous step determines the new input weights. Existence and uniqueness of the limit is established for arbitrary digraphs. Applications to the evaluation of, e.g., sport competitions and paired comparison experiments are discussed.     

Economic Equilibrium: Optimality and Price Decentralization

Mathematical economics has a long history and covers many interdisciplinary areas between mathematics and economics. At its center lies the theory of market equilibrium. The purpose of this expository article is to introduce mathematicians to price decentralization in general equilibrium theory. In particular, it concentrates on the role of positivity in the theory of convex economic analysis and the role of normal cones in the theory of non-convex economies.     

Information patterns and Nash equilibria in extensive games: 1

In this paper we explore the relation between information patterns and Nash Equilibria in extensive games. By information we mean what players know about moves made by others, as well as by chance. For the most part we confine ourselves to pure strategies. But in Section 2 behavioral strategies are also examined. It turns out that they can be modeled as pure strategies of an appropriately enlarged game. Our results, applied to the enlarged game, can then be reinterpreted in terms of the behavioral strategies of the original game.