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To extend the analysis of continuous-time general-equilibrium macro models we study 2 parameter variants Lp,q of the Lebesgue spaces, thus gaining separate control on the asymptotic behaviour (p) and the local behaviour (q): they behave w.r.t. p like the spaces ℓp and w.r.t. q like the spaces Lq on a probability space. Such spaces might naturally contain equilibrium variables (paths) as well as time-dependent policies of a macro model. Convolution behaves very well on those spaces, which can be used as a basis for the classical “comparative statics” (see e.g. Mertens and Rubinchik (2011)). Finally, we generalise the classical implicit function theorem (ift) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms.

ffective supply chain management relies on information integration and implementation of best practice techniques across the chain. Supply chains are examples of complex multi-stage systems with temporal and causal interrelations, operating multi-input and multi-output production and services under utilization of fixed and variable resources. Acknowledging the lack of system’s view, the need to identify system-wide and individual effects as well as incorporating a coherent set of performance metrics, the recent literature reports on an increasing, but yet limited, number of applications of frontier analysis models (e.g. DEA) for the performance assessment of supply chains or networks. The relevant models in this respect are multi-stage models with various assumptions on the intermediate outputs and inputs, enabling the derivation of metrics for technical and cost efficiencies for the system as well as the autonomous links. This paper reviews the state of the art in network DEA modeling, in particular two-stage models, along with a critical review of the advanced applications that are reported in terms of the consistency of the underlying assumptions and the results derived. Consolidating current work in this range using the unified notations and comparison of the properties of the presented models, the paper is closed with recommendations for future research in terms of both theory and application.

Since the 1970s, birth history data have become widely available, thanks notably to the Demographic and Health Surveys program. Despite their wide availability, these data remain under-exploited. Computation, even of simple indicators (fertility rates, total fertility rates, mean age at childbearing) and their standard errors, is not direct with such data, and other types of analysis (fertility differentials, reconstruction of fertility trends et cetera) may also involve reorganization of data sets and statistical modeling that represent a barrier to the use of birth history data. This paper presents a Stata software module (tfr2) that was prepared to analyze birth history data in a user-friendly and flexible way. It is designed to be used primarily with DHS data, but can also be used easily with birth histories from other sources. Three types of analysis are performed by tfr2: (1) the computation of age-specific fertility rates and TFRs, as well as their standard errors, (2) the reconstruction of fertility trends, and (3) the estimation of fertility differentials (rate ratios).The tfr2 module is composed of two parts: (1) a Stata command to transform birth history data into a table of births and exposure (tabexp), and (2) A Poisson regression model to compute fertility rates, fertility trends and fertility differentials from a table of births and exposure (produced by tabexp). Standard errors are also computed by tfr2. One can obtain tfr2 free of charge. It will work with Stata 10 and more recent versions of Stata.

We study the intraday dynamics of liquidity around price jumps using trades and quotes data sampled at the 2-minute frequency for the 30 constituents of the Dow Jones Industrial Average (DJIA) index. Based on an original event-type methodology, we shed new light on the debate about the liquidity response by market participants to information arrivals. We show that the relationship between jumps and market liquidity depends on the respective dimensions of liquidity. \textbf{First, the increase in ex-ante and ex-post trading costs is statistically significant but the economic impact remains limited.} Second, jumps are driven by a sharp rise in the demand for immediacy, rather than by weak liquidity supply. As such, jumps do not seem to be due to an endogenous deficient provision of market liquidity. Third, there is rather strong evidence of resilience, whatever the liquidity measures under consideration. We also find that liquidity shocks in the effective spread and the number of trades contribute the most to the likelihood of jumps. Finally, examining post-jump price discovery process, we conclude that liquidity is more informative immediately after jumps compared to the case where there is no jump.

Competition, security of supply and sustainability are at the core of EU energy policy. The Commission argues that making the European gas market more competitive (completing the internal gas market) will be instrumental in the pursuit of these objectives. We examine the question through the eyes of existing models of the European gas market. Can model tell us anything on this problem? Do they confirm or infirm the analysis of the Commission appearing in fundamental documents such the Green Paper, the Sector Inquiry or the new legislation package? We argue that results of existing models contradict a fundamental finding (paragraph 77) of the Sector Inquiry. We further elaborate on the basis of the economic assumption underlying the models, that changing the assumptions implicitly contained in paragraph 77 cast doubts on a large part of the reasoning justifying the completion of the internal gas market. We also explain that models could help arriving at a better definition of the relevant market, which is so important in the reasoning of the Commission. Last we also find model results that question the effectiveness of ownership unbundling. As to security of supply, we explain that models can also contribute to assess the value of additional infrastructure in the context of security of supply, but this potential seems largely untapped. Last we note that sustainability has not yet penetrated models of gas markets. We conclude by suggesting other area of immediate concern, possibly of higher technical difficulty, that modellers could address in future research.

A new alternative in the analysis of manufacturing systems with finite buffers is presented. We propose and study a new approach in order to build tractable phase-type distributions, which are required by state-of the- art analytical models. Called "probability masses fitting" (PMF), the approach is quite simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF shows some interesting properties: it is bounding, monotonic and it conserves the shape of the distribution. After PMF, from the discrete phase-type distributions, state-of-the-art analytical models can be applied. Here, we choose the exactly model the evolution of the system by a Markov chain, and we focus on flow lines. The properties of the global modelling method can be discovered by extending the PMF properties, mainly leading to bounds on the throughput. Finally, the method is shown, by numerical experiments, to compute accurate estimations of the throughput and of various performance measures, reaching accuracy levels of a few tenths of percent.

Recently, Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] characterized extreme inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible nonnegative integer variables (giving the two-row mixed integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We begin by observing that functions giving valid coefficients for the nonnegative integer variables can be constructed by lifting a subset of the integer variables and then applying the fill-in procedure presented in Johnson [23]. We present conditions for these 'general fill-in functions" to be extreme for the two-row mixed integer infinite-group problem. We then show that there exists a unique 'trivial' lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In all other cases (maximal lattice-free triangle with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the case of a triangle with one integer point in the relative interior of each side and non-integral vertices, we present sufficient conditions to yield an extreme inequality for the two-row mixed integer infinite-group problem.

In this paper we address the issue of finding efficient partial equilibria in markets with nonconvexities. This is a problem that has intrigued generation of economists. Beside its theoretical importance this issue is fundamental in energy markets which do not give the right price signals and incentives to maintain existing and invest in new generating capacity. By considering a competitive environment in which consumers maximize utility independently of other agents actions while suppliers are profit maximizers given other market agents actions, we are able to find efficient prices in markets with non-convexities. Based on this result we propose a design for an energy-only market able to give investors the correct price signals.

Former generation capacity expansion models were formulated as optimization problems. These included a reliability criterion and hence guaranteed security of supply. The situation is different in restructured markets where investments need to be incentivised by the margin resulting from electricity sales after accounting for fuel costs. The situation is further complicated by the payments and charges on the carbon market. We formulate an equilibrium model of the electricity sector with both investments and operations. Electricity prices are set at the fuel cost of the last operating unit when there is no curtailment, and at some regulated price cap when there is curtailment. There is a CO2 market and different policies for allocating allowances. Todays situation is quite risky for investors. Fuel prices are more volatile than ever; the total amount of CO2 allowances and the allocation method will only be known after investments has been decided. The equilibrium model is thus one under uncertainty. Agents can be risk neutral or risk averse. We model risk aversion through a CVaR of the net margin of the industry. The CvaR induces a risk neutral probability according to which investors value their plants. The model is formulated as a complementarity problem (including the CVaR valuation of investment). An illustration is provided on a small problem that captures the essence of today electricity world: a choice restricted to coal and gas, a peaky load curve because of wind penetration, uncertain fuel prices and an evolving carbon market (EU-ETS). We show that we might have problem of security of supply if we do not implement a capacity market.

We propose a tactical model which integrates the replenishment decision in inventory management, the allocation of products to warehousing systems and the assignment of products to storage locations in warehousing management. The purpose of this article is to analyse the value of integrating warehouse and inventory decisions. This is achieved by proposing two methods for solving this tactical integrated model which differ in the level of integration of the inventory and warehousing decisions. A computational analysis is performed on a real world database and using multiple scenarios differing by the warehouse capacity limits. Our observation is that the total cost of the inventory and warehousing systems can be reduced drastically by taking into account the warehouse capacity restrictions in the inventory planning decisions, in an aggregate way. Moreover additional inventory and warehouse savings can be achieved by using more sophisticated integration methods for inventory and warehousing decisions.

Multiple class queueing models arise in situations where some flexibility is sought through pooling of demands for different services. Earlier research has shown that most of the benefits of flexibility can be obtained with only a small proportion of cross-trained operators. Predicting the performance of a system with different types of demands and operator pools with different skills is very difficult. We present an approximation method that is based on equivalent loss systems. We successively develop approximations for the waiting probability, the average waiting time and the service level. Our approximations are validated using a series of simulations. Along the way we present some interesting insights into some similarities between queueing systems and equivalent loss systems that have to our knowledge never been reported in the literature.

In this article we analyze firms investment incentives in liberalized electricity markets. Since electricity is economically non storable, it is optimal for firms to invest in a differentiated portfolio of technologies in order to serve strongly fluctuating demand. Prior to the Liberalization of electricity markets, for regulated monopolists, optimal investment and pricing strategies haven been analyzed in the peak load pricing literature (compare Crew and Kleindorder (1986)). In restructured electricity markets regulated monopolistic generators have often been replaced by competing and potentially strategic firms. This article aims to respond to the changed reality and model investment decisions of strategic firms in those markets. We derive equilibrium investment for strategic firms and compare to the benchmark cases of perfect competition and monopoly outcomes. We find that strategic firms have an incentive to overinvest in base-load technologies but choose total capacities too low from a welfare point of view. By fitting the framework to a specific electricity market (Germany) we are able to empirically analyze Investment choices of strategic firms, and quantify the potential for market power and its impact on generation portfolios in restructured electricity markets in the long run.

In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the latticefree convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that [log2(l)] is a lower bound on the split rank of the intersection cut, where l is the number of integer points lying on the boundary of the restricted lattice-free set satisfying the condition that no two points lie on the same facet of the restricted lattice-free set. The use of this result is illustrated to obtain a lower bound of [log2(n +1)] on the split rank of n-row mixing inequalities.

In this paper we develop several polynomial-time interior-point methods (IPM) for solving nonlinear primal-dual conic optimization problem. We assume that the barriers for the primal and the dual cone are not conjugate. This broken symmetry does not allow to apply the standard primal-dual IPM. However, we show that in this situation it is also possible to develop very efficient optimization methods, which satisfy all desired qualities, including the infeasible-start features. Our technique is based on asymmetric primal-dual barrier augmented by squared residual of the primal-dual linear system.

Nonnegative Matrix Factorization (NMF) is a data analysis technique which allows compression and interpretation of nonnegative data. NMF became widely studied after the publication of the seminal paper by Lee and Seung (Learning the Parts of Objects by Nonnegative Matrix Factorization, Nature, 1999, vol. 401, pp. 788--791), which introduced an algorithm based on Multiplicative Updates (MU). More recently, another class of methods called Hierarchical Alternating Least Squares (HALS) was introduced that seems to be much more efficient in practice. In this paper, we consider the problem of approximating a not necessarily nonnegative matrix with the product of two nonnegative matrices, which we refer to as Nonnegative Factorization~(NF)~; this is the subproblem that HALS methods implicitly try to solve at each iteration. We prove that NF is NP-hard for any fixed factorization rank, using a reduction to the maximum edge biclique problem. We also generalize the multiplicative updates to NF, which allows us to shed some light on the differences between the MU and HALS algorithms for NMF and give an explanation for the better performance of HALS. Finally, we link stationary points of NF with feasible solutions of the biclique problem to obtain a new type of biclique finding algorithm (based on MU) whose iterations have an algorithmic complexity proportional to the number of edges in the graph, and show that it performs better than comparable existing methods.

The progressive relocation of part of the Energy Intensive Industries (EIIs) out of Europe is one of the possible consequences of the combination of emission charges and higher electricity prices entailed by the EU-Emission Trading Scheme (EU-ETS). In order to mitigate this effect, EIIs have asked for special power contracts whereby they would be supplied from dedicated power capacities at average (capacity, fuel, transmission and emission allowance) costs. We model this situation on a prototype power system calibrated on four countries of Central Western Europe. In order to capture the main feature of EIIs' demand, we separate the consumer market in two segments: EIIs and the rest. EIIs buy electricity at average cost price while the rest pays marginal cost. We consider two different types of EIIs' contractual arrangements: a single region wide and zonal average cost prices. We also analyze the cases where generators only rely on existing capacities or can invest in new ones. We find that these average cost contracts can indeed partially mitigate the incentive to relocate activities but with quite diverse regional impacts depending on different national power policies. Models are formulated as a non-monotone complementarity problems with endogenous energy, transmission and allowance prices and are implemented in GAMS.

In this paper, we present a bounding methodology that allows to compute a tight lower bound on the cycle time of fork--join queueing networks with blocking and with general service time distributions. The methodology relies on two ideas. First, probability masses fitting (PMF) discretizes the service time distributions so that the evolution of the modified network can be modelled by a Markov chain. The PMF discretization is simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval. Second, we take advantage of the concept of critical path, i.e. the sequence of jobs that covers a sample run. We show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a lower bound on the original cycle time. The tightness of the bound is shown on computational experiments. Finally, we discuss the extension to split--and--merge networks and approximate estimations of the cycle time.

In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function of the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.

We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that for general matrices these bounds are within a factor 1/n of the exact value, where n is the size of the matrices. Moreover, for sets of matrices with independent column uncertainties of with independent row uncertainties, the corresponding bounds coincide with the joint spectral radius. In these cases, the joint spectral radius is also given by the largest spectral radius of the matrices in the set. As a byproduct of these results, we propose a polynomial-time technique for solving Boolean optimization problems related to the spectral radius. We also consider economics and engineering applications of our results which were never considered practice due to their intrinsic computational complexity.