Robust Transmission Network Expansion Planning in Energy Systems: Improving Computational Performance Recent advances about the problem of transmission network expansion planning propose the use of robust optimization techniques, as an alternative to stochastic mathematical programming methods, to make the problem tractable in realistic systems. They consider different sources of uncertainty, mainly related to the capacity and availability of generation facilities and demands, and make use of adaptive robust optimization models. The resulting formulations materialize on three-level mixed-integer optimization problems, which are solved using different strategies. Although it is true that these robust methods are more efficient than their stochastic counterparts, it is also true that solution times for mixed-integer linear programming problems growth exponentially with respect to the size of the problem. This fact encourage researchers, practitioners and system operators to use computational efficient methods when solving this type of problems. This paper addresses the issue of improving computational performance by taking different features from existing algorithms. In particular, we replace the lower-level problem by its dual, and solve the resulting bi-level problem using a primal cutting plane algorithm within a decomposition scheme. Using this alternative and simple approach, the computing time for solving transmission expansion planning problems reduces drastically. Numerical results on an illustrative example, the IEEE-24 and IEEE 118-bus test systems demonstrate that the algorithm * Corresponding author: rominsol@gmail.com, tlfn.: 00 34 926810046January 23, 2015 is superior in terms of computational performance with respect to existing methods. The problem of Transmission Network Expansion Planning (TNEP) aims to resolve the issue of how to expand or reinforce an existing electricity transmission network so that adequately serves system loads over a given horizon. The main difficulty of this problem is to take decisions under the great amount of uncertainty associated with i) demands, ii) renewable generation, such as wind and solar power plants, and iii) equipment failure. Two main frameworks have been used to tackle these uncertainties, stochastic programming [1] and robust optimization [2, 3, 4, 5, 6, 7, 8] .. In stochastic programming problems, uncertain data is assumed to follow a given probability distribution and are usually dealt with by using scenarios or finite sampling from the joint probability density funtion. Examples of the successful application of stochastic programming in TNEP problems are given by [9, 10, 11, 12] among others. However, the number of scenarios needed to represent the actual stochastic processes can be very large, which may result in intractable problems. In contrast, robust optimization can avoid the intractability issue related to stochastic programming approaches. Note that tractability is not the only difficulty arising from stochastic programming, but it is the most important aspect from this paper perspective. Examples of the successful application of robust optimization in TNEP problems are given by [13, 14] . However, in this paper we focus on the seminal works proposed by [15] , [16] and [17] .. The three works on which our proposal is based formulate the problem using an adaptive robust optimization (ARO) framework [18, 19] . In consequence, all of them use three-level formulations: i) first level minimizes the cost of expansion ([16] also minimizes the maximum regret), the decision variables for this level are those associated with construction or expansion of lines, ii) second level selects the realization of the uncertain parameters that maximizes the system's operating costs within the uncertainty set, the variables related to this level are the uncertain parameters, i.e., generation capacity and demand, and iii) in the third level the system operator selects the optimal decision variables to minimize operating costs for given values of first and second level variables. Note that in this work we do not consider contingencies explicitly, however, they could be taken into account within the robust formulation following the guidelines proposed by [20] . [15] transforms the second and third levels into one single-level maximization problem (so called subproblem in this work) by using the dual of the third level. It takes advantage of the fact that all uncertain parameters need to be equal to their upper or lower limits in the worst case scenario, which allows using binary variables to deal with these uncertain parameters. This simplification imposes the limitation that the uncertainty budget, typically used in robust optimization, must be integer. However, in our opinion, it does not dwarf the benefits of robust optimization while simplifying the resolution of the problem substantially. In contrast, [16, 17] transforms the second and third levels into one single level maximization problem by using the Karush-Kuhn-Tucker (KKT) conditions of the third level. In this particular case, the number of constraints, continuous and binary variables of the subproblem increases with respect to the other approach.. Regarding the resolution of the resulting bi-level problem, [15] proposes a Benders [21] approach where the dual information from subproblems is used to construct additional Benders cuts. The main concern of this approach is the slow convergence typical from this type of decomposition algorithms, which forced the author to include additional linear constraints in order to improve convergence. On the other hand, [16, 17] apply a constraint-andcolumn [22] generation method solely based on primal cuts. This method is computationally advantageous with respect to the Benders decomposition and converges in a small number of iterations (authors reported solution results with only 3-4 iterations required). Note that the methodologies proposed by [16] and [17] are basically the same from the mathematical point of view, however, they analyse different situations and [17] proposed a different way to construct the uncertainty sets.. This paper proposes an alternative solution approach to the three-level mixed-integer optimization problem associated with TNEP. The idea is to combine the way [15] solves the inner level problems (subproblems) and the constraint-and-column generation method used by [16] and [17] for the first level problem (master problem). Note that although the master problem solved using primal cuts is very efficient, computing times associated with [16] and [17] approaches increase exponentially with respect the size of the problem. The reason is the subproblem, which takes most of the resolution time because of the use of the KKT optimality conditions of the third level into the second level problem. The resulting model calculates simultaneously the primal and dual variables of the third level problem, and the primal variables are not required to construct the primal cuts of the master. Therefore, this alternative of using the KKT conditions is inefficient.. The remainder of the paper is structured as follows. Section 2 describes the adaptive robust formulation of the TNEP problem in compact matrix form, and discusses the solution approaches proposed by [15] , [16] and [17] . The detailed formulation is only given in the Appendix A. In Section 3 the definition of the uncertainty set and the description of the solution approach are presented. Numerical results for different networks are given in Section 4 and compared with those obtained using the method proposed by [16] and [17] . Finally, the paper is concluded in Section 5. This paper proposes a new decomposition algorithm to attain the exact solution of the TNEP problem derived from using a two-stage adaptive robust strategy. Although there is nothing novel with respect the formulation of the problem, we manage to combine formulations and findings from different authors in the proper manner so that our method beats in computational efficiency, same cases by large, prior methodologies.. We have run several computational experiments to show the functioning of the proposed algorithm with respect to the approach proposed by [16] and [17] . Nevertheless, it has been demonstrated that the computational time required by this algorithm is always lower than those proposed by [15] , [16] and [17] . The subproblem proposed in this paper has lower complexity and number of binary variables than the others, while the primal master problem is equal to the one proposed by [16] and [17] , which is computationally advantageous with respect to the Benders master problem used by [15] , and besides, it is not required to include additional cuts from given realizations of the random parameters.. Bearing in mind that the proposed method keeps all features about robust optimization design with respect to their predecessors, the proposed algorithm turns out to be the most efficient method to solve the robust transmission network expansion planning to date, and takes advantage of state-of-the-art mixed-integer mathematical programming solvers. The only limitation is the integrality constraint of budget uncertainty, which simplifies the subproblem considerably and does not dwarf the benefit of using robust optimization. Numerical experiments demostrate that the proposed method is computationally more efficient than that proposed by [16] and [17] . Nevertheless, this result is evident considering the difference in complexity of the subproblem. Note that [17] already claimed that the most time consuming stage of their process was the subproblem resolution, and our subproblems make a difference in terms of saving computational time while master problems are exactly the same.. We have not reported results with respect to [15] method. Nevertheless, it is also evident that our method is more efficient because we use the same subproblem but reducing binary variables to a half. This difference might imply large computing time differences for large problems. In fact, the larger the problem, the larger the difference according to the functioning of standard branch-and-cut solvers. The only doubt about if our method is faster might come from the master problem, however, the number of iterations reported in this paper and in the works by [16] and [17] are always lower than or equal to 5, and the number of iterations reported by [15] varies considerably depending if additional cuts are included or not but they are usually above 5 or more iterations. Besides, this strategy of including additional cuts could be reproduced using the primal master problem of our algorithm by just considering additional realizations of random parameters like [15] does in his work. Nevertheless, the convergence behavior does not require it.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


FREQUENCY-RESPONSE MASKING BASED FIR FILTER DESIGN WITH POWER-OF-TWO COEFFICIENTS AND SUBOPTIMUM PWR This paper presents a new method for designing sharp linear phase FIR filters with power-of-two coefficients. The method is based on a frequency-response masking technique. In this method, the power-of-two coefficients and continuous scaling parameters of the subfilters are taken to be decision variables, and minimizing peak weighted ripple (PWR) is taken to be the design objective. The resulting nonlinear mixed integer optimization problem for each subfilter is first reduced to an equivalent discrete optimization problem whose search region is then cropped for efficiency of computation, similar to the approach in Ref. 1, although a different cropping strategy is used here. The effectiveness of the method is demonstrated through a lowpass linear phase sharp FIR digital filter example. It is demonstrated in Refs. 2-4 that the frequency-response masking (FRM) technique is an efficient method for reducing the complexity of a sharp FIR filter. The basic structure of a filter synthesized using the FRM technique is shown in Fig. 1 . If F (z) is the z-transform transfer function of the system, then, from Fig. 1 ,. Here, F a (z M ) is obtained by replacing each delay element of a prototype filter, F a (z), by M delay elements. F c (z M ) is the complement of F a (z M ), and F M a (z), F M c (z) are the masking filters of the system. The frequency-response magnitudes, |F a (e jM w )|, |F c (e jM w )|, |F M a (e jw )| and |F M c (e jw )|, of these filters are shown in Fig. 2 . If F a (z M )F M a (z) and F c (z M )F M c (z) have the same phase delay, the resulting frequency-response magnitude, |F (e jw )|, of F (z) is shown in Fig. 2(c) . The transition width of F (z) is narrower than that of F a (z) by a factor of M . The complexity of the filter can be further reduced by restricting the coefficients for each subfilter to lie in a power-of-two space. A new method is developed for designing a FRM based FIR digital filter in the power-of-two space. Numerical results obtained from the example show that our method can yield quality solutions with modest computational effort.

B I O O B I O B I O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Spatially-resolved metabolic cooperativity within dense bacterial colonies Background: The exchange of metabolites and the reprogramming of metabolism in response to shifting microenvironmental conditions can drive subpopulations of cells within colonies toward divergent behaviors. Understanding the interactions of these subpopulations-their potential for competition as well as cooperation-requires both a metabolic model capable of accounting for a wide range of environmental conditions, and a detailed dynamic description of the cells' shared extracellular space. Results: Here we show that a cell's position within an in silico Escherichia coli colony grown on glucose minimal agar can drastically affect its metabolism: "pioneer" cells at the outer edge engage in rapid growth that expands the colony, while dormant cells in the interior separate two spatially distinct subpopulations linked by a cooperative form of acetate crossfeeding that has so far gone unnoticed. Our hybrid simulation technique integrates 3D reaction-diffusion modeling with genome-scale flux balance analysis (FBA) to describe the position-dependent metabolism and growth of cells within a colony. Our results are supported by imaging experiments involving strains of fluorescently-labeled E. coli. The spatial patterns of fluorescence within these experimental colonies identify cells with upregulated genes associated with acetate crossfeeding and are in excellent agreement with the predictions. Furthermore, the height-to-width ratios of both the experimental and simulated colonies are in good agreement over a growth period of 48 hours. Conclusions: Our modeling paradigm can accurately reproduce a number of known features of E. coli colony growth, as well as predict a novel one that had until now gone unrecognized. The acetate crossfeeding we see has a direct analogue in a form of lactate crossfeeding observed in certain forms of cancer, and we anticipate future application of our methodology to models of tissues and tumors. The application of the 3DdFBA method described here focused on the growth and collective metabolism of E. coli colonies. Its integration of 3D reaction-diffusion simulations with a flux balance model of metabolism that involves thousands of reactions and metabolites enabled us to generate new hypotheses that we then tested directly in the laboratory.. The most striking result of this work was the prediction that subpopulations within E. coli colonies naturally engage in cooperative acetate crossfeeding. This was not due to the evolution of distinct crossfeeding genotypes, as is known to occur in long-term continuous culture experiments, but rather it emerged from the cells' own regulatory responses to their local microenvironments within the colony. Depending on location, some cells experienced a glucose-rich anoxic environment that drove them toward acetate-producing fermentative metabolism, while others experienced a glucose-poor aerobic environment that favored acetate-consumption. This behavior remained robust over a range of common agar glucose concentrations, meaning that it may be occurringcompletely unnoticed-in laboratory incubators the world over.. The simulated colonies exhibited realistic growth dynamics. The same glucose and oxygen gradients that gave rise to acetate crossfeeding also gave rise to a ring of fast-growing pioneer cells at the colony's edge, and significantly hindered the growth of much of the colony interior. The pioneer ring had a profound impact on the macroscopic shape and growth of the simulated colony, leading to both its broad and flat appearance and its linear radial expansion (both of which agree well with experimental values).. Several features of our modeling technique proved essential to our study. The first, and most critical, was its ability to perform 3D simulations with fine spatial resolution. The concentration profiles of oxygen and glucose within the colony changed dramatically over short distances in both the radial and vertical directions. The penetration depth of oxygen, for example, was only around 50 μm from the edge of the colony, while the penetration depth of glucose at the colony center was only around 60 μm upward from the agar. Accurately accounting for these steep gradients required the use of a 3D lattice with a spacing of on the order of 10 μm. Although a similar method was recently reported [21] , it was restricted to two spatial dimensions and its reliance on costly onthe-fly FBA calculations severely limit its practicality for performing the millions of metabolism evaluations pertimestep necessary to resolve these profiles. In contrast, our use of precomputed FBA lookup tables enabled us to preform our simulations at nearly real-time speeds. Our code, running on the Keeneland supercomputer (with Nvidia M2090 GPUs) and on a Linux desktop machine (with a single Nvidia GTX780 GPU) performed the simulations presented here at speeds of approximately 50.0 and 40.1 simulated minutes per wall-clock hour, respectively.. The second feature that proved critical was our method's ability to account for the regulation of resource use by the modeled cells. Naively using FBA alone led to some cells simultaneously taking up glucose and acetate in a manner at odds with experimental data. By requiring the colony to obey a phenomenological model of the acetate switch, this pathological behavior was ameliorated, and the potential for acetate crossfeeding to emerge within wild-type E. coli colonies was more realistically modeled.. It is worth noting that our use of precomputed FBA tables, while offering vast speedup over on-the-fly evaluation, does carry some drawbacks. The most pressing of these is that the modeler must have some notion of the substrates that are likely to play a role in the colony's collective metabolism before a simulation can be launched. Our choice to model only glucose, oxygen, and acetate was informed by earlier work simulating significantly smaller clusters of cells [10] and some preliminary FBA calculations. Nevertheless, this choice can in some ways limit the scope of the simulations. For example we knowingly ignore the potential for succinate crossfeeding (which was deemed unimportant due to the low predicted succinate production rate of anaerobically-growing cells) as well as possible toxic effects due to the production of ethanol within the colony (although, because the ethanol and acetate production rates are comparable, the concentration of ethanol should not rise significantly above the maximal acetate concentration, ∼ 5.8 × 10 −3 M, which is approximately 2 orders of magnitude below the concentration at which cellular growth is significantly inhibited [46] ). Incorporating larger numbers of substrates into a lookup table is straightforward, but can be time consuming as it increases the dimensionality of the table, and in turn, the number of FBA calculations required to produce it. Ultimately, the choice of using table lookups or on-the-fly FBA solving comes down to a choice between computational speed/resolution and model flexibility/universality. With enough prior knowledge of the phenomenon to be simulated, a table-based method vastly out-performs on-the-fly solving, but for purely exploratory simulations in which large numbers of metabolites are simulated simultaneously, an on-thefly method may potentially yield novel behaviors that the modeler did not anticipate.. Although our methodology represents an important step in using FBA in both a spatially-and temporallyresolved manner, neither it, nor any other current implementation, can fully account for all biologically significant phenomena. One important example stems from the inherent determinism of the method which yields only average behaviors. Stochastic gene expression has been shown to give rise to significant metabolic variability, even among cells in otherwise identical environments [6] . The method we present does not account for this type of cell-to-cell variability; the cells in a given lattice site are assumed to behave identically, engaging in the optimal metabolism possible given their local substrate availability. Additional uncertainties arise in the number of cells that actually seed a given colony; our simulations assume a single cell, but it is difficult to experimentally verify the exact number of cells left in a given location when an agar plate is streaked. These effects are evident even at the macroscopic level-while the heights and widths of experimental colonies show considerable variability (see Figure 6 ), our simulated colonies show none. A natural way forward in this regard is to shift toward an agentbased modeling approach where individual cells may be represented within a continuous field of diffusing substrates. This could allow for stochastic gene expression to be explicitly accounted for within each cell; the gene expression state of a cell might be sampled from experimental distributions and used to apply constraints within the cell's metabolic network in a manner similar to that of [6] . Additionally, by resolving individual cells and their intercellular forces, an agent-based approach may better account for the early development of the colony as it transitions from 2D to 3D growth [44] -a behavior that is poorly accounted for in our current implementation.. Our modeling technique can be applied to the study of communities involving many different microbial species. Several different cell types-each employing their own metabolic model-can be simultaneously simulated, and because the different cell types can transform into each other, regulation systems much more complex than the acetate switch described here can be studied. Beyond obvious future studies of biofilms, perhaps the most exciting applications of this technique may come in the form of tissue and tumor modeling. Like the colony models presented here, steep oxygen gradients are known to form within cancerous tumors that profoundly affect their metabolism [47] . Within certain cancers these oxygen gradients have even been reported to give rise to lactate crossfeeding in a manner strikingly similar to the acetate crossfeeding seen in our simulations [48, 49] . There are already several flux balance models of different human cancers available in the literature [50, 51] . These can be leveraged to build new 3D models capable of studying everything from environmental fluctuations and metabolic reprogramming within a cancer to the interactions between cancers and their surrounding tissues.

O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Priority Levels Based Multi-hop Broadcasting Method for Vehicular Ad hoc Networks Abstract This paper deals with broadcasting problem in Vehicular Ad hoc Networks (VANETs). This communication mode is commonly used for sending safety messages and traffic information. However designing an efficient broadcasting protocol is hard to achieve since it has to take into account some parameters related to the network environment, for example the network density, in order to avoid causing radio interferences. In this paper, we propose a novel Autonomic Dissemination Method (ADM) which delivers messages in accordance with given priority and density levels. The proposed approach is based on two steps: an offline optimization process and an adaptation to the network characteristics. The first step uses a genetic algorithm to find solutions that fit the network context. The second one relies on the Autonomic Computing paradigm. ADM allows each vehicle to dynamically adapt its broadcasting strategy not only with respect to the network density, but also in accordance to the priority level of the message to send. The experimental results show that ADM effectively uses the radio resources even when there are globally many messages to send simultaneously. Moreover, ADM allows to increase the message delivery ratio and to reduce the latency and radio interferences. A Vehicular Ad hoc Network (VANET) is a collection of vehicles communicating through wireless connections: each vehicle acts simultaneously as a node and as a wireless router, allowing multi-hop packet forwarding. Indeed, each node has a limited coverage area that contains the neighbours it can directly communicate with. This area can vary from one hundred meters to a few kilometers (depending on the Wahabou Abdou · Benoît Darties · Nader Mbarek LE2I -UMR CNRS 6306, University of Burgundy, 9 Avenue Alain Savary, 21 000 Dijon France E-mail: {wahabou.abdou, benoit.darties, nader.mbarek}@u-bourgogne.fr wireless technology on-board, external radio interferences,. . . ). Using vehicle-tovehicle communications allows sending packets over wide distance through multihop relays. VANET are mainly characterized by a dynamic network topology and a heterogeneous node density due to road traffic conditions. This paper deals with broadcasting techniques which are used for sending safety messages, traffic information or comfort messages. When a packet is broadcasted, it is received by all nodes within the sender's coverage area (provided that no interference or radio channel trouble occurs). Every receiver will decide to relay or not the packet depending on its own broadcasting strategy. This hop-to-hop communication would lead to a full coverage of the network. Performing an efficient multi-hop broadcast in VANETs is however a difficult task. The protocols should take into account the specificities of the radio channel, the high mobility of nodes and the network density. The decision to relay the packets is taken in a distributed way, but each node's decision has a real impact on the efficiency of the overall dissemination strategy: in high-density networks, too many relay nodes would quickly increase the number of collisions, leading to a saturation of the bandwidth and a significant increase of the latency. On the other hand, if not enough relay tasks are performed in low-density networks the message may not be widely disseminated. Several approaches are proposed in the literature to overcome this problem. Some methods from Mobile Ad hoc Networks (MANETs) use the neighborhood knowledge [13] [11] to choose the best relay nodes. However, the high mobility that characterises VANETs makes their application in such a network very difficult. Some stochastic methods use waiting time to reduce the number of redundant packets [8] [14] [10] [4] [6] . But if this waiting time is not well chosen, it may increase the message dissemination time. The approach we present in this paper is based on the Smart-flooding protocol [1] that uses a genetic algorithm optimization process to dynamically adapt dissemination strategies with respect to the network density.. It is also important to adapt the broadcasting strategy to the priority level of the message. For instance, emergency messages such accident alerts, should be delivered as fast as can be done in the source node's neighbourhood. Conversely, it does not matter if some weather information with limited impact and less urgent tourist information are broadcasted with a more important latency since there is no emergency. This paper investigates the problem of building an autonomous and robust broadcasting protocol, which provides each node with the adequate strategy to determine if an incoming message has to be forwarded or not depending on its priority level and the network density. The goal is to make effective use of radio resources when there are many messages to send simultaneously. The paper is organized as follows: Section 2 discusses a state of the art of existing broadcasting methods. Thereafter, Section 3 formulates the broadcasting problem in VANETs as a multiobjective problem and presents an optimization methodology. The proposed Autonomic Dissemination Method (ADM) is detailed in Section 4 and its performances are assessed in Section 5. The P , N r Dr and T T L parameters are optimized using HOPES (Hybrid Optimization Platform using Evolutionary Algorithms and Simulations). Our platform combines an optimizer, a network simulator and a trace analyzer. Figure 1 illustrates the interaction of these three modules.. The Optimization Engine is used to effectively explore the search space while the Network Simulator assesses solution using models that are closed to reality. We used aGAME(adaptive Genetic Algorithm with Multiple parEto sets) [2] as optimizer tool. aGAME generates a set of possible solutions. Thereafter, theses solutions are transmitted to the network simulator which integrates them with other parameters in order to better reproduce the context of a real network. The trace files generated during the simulation are then transmitted to the Trace Analyzer module. The latter processes the trace files in order to extract the values of the objective functions (NC, P T , R, and F R) and presents the obtained results according to format required by the genetic algorithm. Then, aGAME uses these results to guide the exploration of the search space. The process is repeated until a stop criterion is met (for instance the total number of solutions to evaluate).. The overall optimization process leads to a set of solutions, corresponding to broadcasting strategies that fit a network with a given density level. This process is repeated for several densities by changing the corresponding parameter in the Network Simulator module. It is worth mentioning that this is an offline optimiza-tion process. The results (that represents the "best-suited" broadcasting strategy for each density) allow building a knowledge base that establishes a connection between density levels and broadcasting strategies. Each vehicle can therefore choose the appropriate dissemination strategy, depending on the density of the network in which it is located. The evaluation of the density level is discussed in section 4.2.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Step Forward in Studying the Compact Genetic Algorithm The compact Genetic Algorithm (cGA) is an Estimation of Distribution Algorithm that generates offspring population according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The cGA only needs a small amount of memory; therefore, it may be quite useful in memory-constrained applications. This paper introduces a theoretical framework for studying the cGA from the convergence point of view in which, we model the cGA by a Markov process and approximate its behavior using an Ordinary Differential Equation (ODE). Then, we prove that the corresponding ODE converges to local optima and stays there. Consequently, we conclude that the cGA will converge to the local optima of the function to be optimized. One of the most famous optimization procedures for combinatorial optimization is the Genetic Algorithm (GA). By maintaining a population of solutions, the GA can be viewed as implicitly modeling of the solutions seen in the search process. In the standard GA, new solutions are generated by applying randomized recombination operators on two or more high-quality individuals of the current population (Goldberg, 1989 ). These recombination operators, such as one-point, two-point or uniform crossover, randomly select non-overlapping subsets of two "parent" solutions to form "children" solutions. By using a crossover operator that preserves groups of parameters from parents to children, the GA attempts to capture dependencies between the parameters implicitly.. The poor behavior of genetic algorithms in some problems, sometimes attributed to designed operators, has led to the development of other types of algorithms. The Probabilistic Model Building Genetic Algorithms (PMBGAs) or Estimation of Distribution Algorithms (EDAs) are a class of algorithms which has been developed recently to preserve the building blocks (Larranaga and Lozano, 2001) . The principal concept in this new technique is to prevent the disruption of partial solutions contained in a solution by building a probabilistic model. The EDAs are classified into three classes based on the interdependencies between the variables of solutions (Larranaga and Lozano, 2001; Pelikan et al., 1999) ; the no dependencies model, the bivariate dependencies model, and the multiple dependencies model. To name just a few, instances of EDA algorithms include the Population-based Incremental Learning (PBIL) (Baluja, 1994; Baluja and Caruana, 1995) , the Bit-based Simulated Crossover (BSC) (Syswerda, Parameters: α is the learning step, n is the solution length. Step 1. Set k to 0, and initialize the probability vector For i := 1 to n do p i (k) := 0.5; Step 2. Generate two solutions from the probability vector a(k) := generate(p(k)); b(k) :=generate(p(k)); Step 3. Let them compete w(k), l(k) := compete (a(k), b(k)); where w(k) and l(k) are winner and loser solutions respectively. (if both a(k) and b(k) have the same fitness value then a(k) is selected as w(k)) Step 4. Update the probability vector. Step 5. Check if the probability vector has converged.. Go to Step 2, if it is not satisfied. Section 4, the analysis of the cGA as a Markov process is done in two stages. In the first stage, we derive an ODE whose solution approximates the asymptotic behavior of the cGA. Then in the second stage, we prove that the corresponding ODE and therefore, the cGA surely converge to the local optima of the function to be optimized and stays at them. Finally, Section 5 concludes the paper. The cGA is an estimation of distribution algorithm. It is very simple and can be easily implemented in hardware. Using a small amount of memory, it can have many applications in the memory constraint problems. In this paper, a mathematical framework of the cGA, based on the weak convergence and the non-linear systems theories, has been proposed and consequently, the convergence behavior of the cGA has been studied. We have proven that the local maxima of an injective function are asymptotically stable stationary points of the cGA and shown that the cGA converges to one of these local maxima. While the results obtained in this paper are interesting by their own, they can also serve as one of the first steps towards using ODE in analysis of EDAs and the other evolutionary algorithms.. There are a lot of open questions and we have planned to study them in the future. First of all, we are interested in extending our framework to non-injective functions and determining the convergence rate of the cGA for different functions. Recently, some theorems have been developed in the stochastic approximation theory that can be useful in this regard (Kushner and Yin, 2000) . Since for each optimization algorithm chosen to solve a problem, the shape and size of the basin of attractions may be different, we also would like to compute the basin of attractions of local maxima for a determined function. Comparing the basin of attractions of local maximum for the cGA with the basin of attractions of local maxima for other algorithms help us in choosing a better algorithm for optimization of a determined function. For example if we could show that the basin of attraction of global maximum for the cGA is bigger than the basin of attraction of the same point for the PBIL then we can predict that for different initial values the cGA will converge to the global maxima with a higher probability.

O O O O O O B I I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The list scheduling algorithm for scheduling unreliable jobs on two parallel machines In this paper, we study a scheduling problem with unreliable jobs. Each job is characterized by a success probability and by a reward earned in case of success. In case of failure, the job blocks the machine that is processing it, and the jobs subsequently sequenced on that machine cannot be performed. The objective function is to maximize the expected reward. We address the problem in the case of two parallel machines, and analyze the worst-case performance of a simple list scheduling algorithm. We show that the algorithm provides an approximation ratio of (2 + √ 2)/4 ≃ 0, 853, and that the bound is tight. We also provide a complexity result concerning the related Total Weighted Discounted Completion Time Problem on parallel machines. A set of jobs J = {J 1 , . . . , J n } must be allocated to m parallel, identical machines, M 1 , . . . , M m . Each job must be assigned to a single machine and a machine can process one job at a time. Jobs are unreliable, i.e., while a job is being processed by a machine, a failure can occur, which implies losing all the work that was scheduled but not yet executed by the machine. Each job J i is characterized by a certain success probability π i (independent from other jobs) and a reward r i , which is gained if the job is successfully completed. The problem is to find an allocation of jobs to the m machines and a sequence on each machine that maximizes total expected reward. In the following, we refer to this problem as Unreliable Job scheduling Problem, denoted by UJP. We use UJP(m) to denote UJP with m machines.. Let σ j be a sequence of jobs assigned to machine M j , and let σ j (k) be the job in k-th position in σ j . A feasible solution σ = {σ 1 , σ 2 , . . . , σ m } for UJP is an assignment and sequencing of the n jobs on the m machines. The expected reward associated with sequence σ j is obtained by summing up the contribution of the first, the second, . . . the k-th job in σ j , taking into account that a job can be processed only if no breakdown on that machine occurred so far. Hence, if K jobs are sequenced on a machine M j according to σ j , recalling that the success probabilities are independent, the expected reward. (1) and the total expected reward is therefore. UJP consists in finding a solution σ * = {σ * 1 , σ * 2 , . . . , σ * m } that maximizes the total expected reward. We let z * denote the value of such optimal solution. By a pairwise interchange argument, it can be shown [8] that the single-machine version of UJP is solved by sequencing the jobs in nonincreasing order of the following Z -ratio [8]: . If γ ≪ 1, one has  w i e −γ C i ≈  w i −  w i C i , and therefore TWDCTP tends to become equivalent to the Total Weighted. Completion Time Problem (TWCTP), i.e., the problem of minimizing the weighted sum of job completion times. Such problem is solved by the well-known Smith's rule, i.e., jobs must be processed in nonincreasing order of the ratio. and as γ → 0 in (3), the two rules (3) and (4) . solved by a simple priority index sequencing rule.) Note that all these problems (UJP, TWDCTP, TWCTP) essentially consist in deciding how to partition the n jobs among the m machines, since on each machine the sequencing is then dictated by the priority rule. The plan of the paper is as follows. In Section 2, some results and applications from the literature are reviewed. In Section 3, the relationship between UJP, TWDCTP and TWCTP is investigated. In Section 4, the list scheduling heuristic algorithm for UJP is proposed, and the worst case behavior is analyzed. Finally, conclusions follow.

O B I O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O


Solving the Resource Constrained Project Scheduling Problem Using the Parallel Tabu Search Designed for the CUDA Platform In the paper, a parallel Tabu Search algorithm for the Resource Constrained Project Scheduling Problem is proposed. To deal with this NP-hard combinatorial problem many optimizations have been performed. For example, a resource evaluation algorithm is selected by a heuristic and an effective Tabu List was designed. In addition to that, a capacity-indexed resource evaluation algorithm was proposed and the GPU (Graphics Processing Unit) version uses a homogeneous model to reduce the required communication bandwidth. According to the experiments, the GPU version outperforms the optimized parallel CPU version with respect to the computational time and the quality of solutions. In comparison with other existing heuristics, the proposed solution often gives better quality solutions. The Resource Constrained Project Scheduling Problem (RCPSP), which has a wide range of applications in logistics, manufacturing and project management [1] , is a universal and well-known problem in the operations research domain. The problem can be briefly described using a set of activities and a set of precedence constraints describing the relationships among activities. Each activity requires a defined amount of the resources and every resource has a limited capacity. The objective is to find the best feasible schedule according to a criterion. The RCPSP was proved to be NP-hard in the strong sense when the criterion is makespan [2] . For that reason only small instances (approximately up to 30 activities) can be reliably solved by exact methods like Branch & Bound [3] , therefore a heuristic or a meta-heuristic is required to solve the problem satisfactorily.. In recent times, there is an increased interest in using graphics cards to solve difficult combinatorial problems (e.g. [4, 5, 6] ), since a modern graphics card is usually much more powerful than a current multi-core CPU. Although the graphics cards have some restrictions (e.g. a high-latency global memory access), the new GPU architectures like Kepler and Fermi can significantly reduce these bottlenecks. As a consequence modern GPUs are applicable to the problems which were solvable only on CPUs previously.. Not only the high computational power makes graphics cards attractive to researchers and practitioners, but also the mature Nvidia CUDA framework which enables us to create GPU programs in an effective and relatively easy way since it extends standard languages like C/C++ by adding GPU specific functions and language keywords. Nevertheless, the CUDA is only designed for the Nvidia graphics cards.. From the implementation point of view, there are two models. The first one is called a homogeneous model where all required data structures are stored in a GPU at the beginning of an algorithm and the results are read at the end of the algorithm. There is no communication between the CPU and the GPU during the computations. The second approach is a heterogeneous model. The main logic of an algorithm runs on the CPU and the GPU is used only for the most computationally intensive tasks. The disadvantage of the heterogeneous model is the frequent communication during computations, therefore, the communication bandwidth can state a bottleneck. However, the heterogeneous model is usually simpler to implement. The first known GPU algorithm dealing with the Resource Constrained Project Scheduling Prob-lem has been proposed. The performed experiments on the standard benchmark instances reveal the merits of the proposed solution. The achieved quality of solutions is very good and outperforms the other Tabu Search implementations to the best of our knowledge. In addition to this, the GPU algorithm design has proved to be very effective since the mid-range GPU was substantially faster than the optimized parallel CPU version. The Nvidia Geforce GTX 650 Ti GPU is able to evaluate more than one million schedules per second for the J120 data-set on average. The achieved performance boost could not be reached without effective structures and auxiliary algorithms. The Simple Tabu List implementation is adapted to the features of the GPU, the capacityindexed evaluation algorithm was proposed and many parallel reductions were applied. In addition to this, the homogeneous model reduces the required communication bandwidth between the CPU and the GPU.. In spite of the fact that GPUs are not primarily designed for solving combinatorial problems the rising interest about these solutions can be seen [24] . The reason for this is the high computational power of graphics cards and the relatively user friendly programming API that the CUDA offers. So it can be expected that GPUs will be more and more used in operations research in the future. The Tabu Search meta-heuristic was proposed by Glover in 1986 [7] . Hundreds of publications have been written since that time. The basic concept of the TS meta-heuristic is clarified in Gendreau [8] . The author has described the basic terms of the TS, as a Tabu List (TL), aspiration criteria, diversification, intensification, etc.. From the Tabu Search parallelization point of view James et al. [9] proposed a sophisticated solution. The authors use a circular buffer where the size of the buffer is equivalent to the number of the started threads (often the number of CPUs cores). Every location (i.e. an index of a thread) has different parameters (a tabu tenure, stopping criteria). At the beginning of the search each thread initializes its own location by a short TS operator, i.e. a modified version of the Taillard's robust tabu search. Then the asynchronous parallel tabu search is started. Every thread independently reads a solution and parameters from the location, possibly makes a diversification, runs the TS operator on the solution, and writes back and sets an UPDATE flag if an improving solution is found. After that the thread location index is circularly incremented. Diversification takes place if the read solution does not have the UPDATE flag set. Every best global solution is copied to half of the locations of the circular buffer to propagate elite solutions. Since the circular buffer is shared by many threads, the access has to be as short as possible and the locations have to be protected by critical sections.. Relatively many authors try to use a GPU for solving combinatorial problems. For example, Czapiński and Barnes [10] implemented a GPU version of TS to solve the Flowshop Scheduling Problem (FSP). The success of the implementation illustrates an achieved speedup against the CPU version. The GPU version was up to 89.01 times faster than the CPU (Intel Xeon 3.0 GHz, 2 GB memory, Nvidia Tesla C1060 GPU). Nevertheless, the quality of solutions was not investigated.. The Flowshop Scheduling Problem was also solved by Zajíček andŠůcha [11] . The authors implemented a GPU version of an island based genetic algorithm. Islands are used for migration of individuals among sub-populations, where each sub-population is a subset of solutions and an individual corresponds to a specific solution. Subpopulation can be evaluated, mutated and crossed over independently of other sub-populations, therefore, huge parallelization can be achieved. It should be noted that a homogeneous model was used. The maximal speedup against the CPU was 110 for 100 activities and 5 machines (AMD Phenom II X4 945 3.0 GHz, Nvidia Tesla C1060).. Czapiński [5] proposed a Parallel Multi-start Tabu Search for the Quadratic Assignment Prob-2 lem. The main idea is to start several Parallel Tabu Search instances with different parameters and initial solutions. All Tabu Search instances should terminate approximately at the same time since the synchronization is required to get the most promising solutions. When a stop criterion is met, the modified solutions are read back and the most promising solutions are used as the initial solutions in the next run. The Tabu Search instance runs entirely on the GPU, therefore communication overheads are reduced to minimum. The achieved results reveal the effectiveness of the implementation since Nvidia GTX 480 is up to 70 times faster than a six-core Intel Core i7-980x 3.33 GHz.. Hofmann et al. [12] investigated the suitability of graphics cards for genetic algorithms. The authors selected two problems to solve, namely the Weierstrass function minimization and the Traveling Salesman Problem. The first problem can be very effectively implemented on the GPU since the Weierstrass function is comprised of floatingpoint operations and trigonometric functions that are directly supported by the GPU hardware. In contrast to the first problem the second problem was not tailor-made for the GPU, therefore, the multi-core CPU was able to compete with the GPU. The authors suggest that all parts of a genetic algorithm should be performed on the Fermi or newer GPUs (i.e. homogeneous model).. Boyer et al. [13] used dynamic programming to solve the knapsack problem on a GPU. An effective data compression was proposed to reduce memory occupancy. The achieved results show that the Nvidia GTX 260 graphics card was up to 26 faster than Intel Xeon 3.0 GHz.. The above mentioned combinatorial problems have something in common. The solution evaluation is quite simple since it is usually a "simple sum". On the other hand, the RCPSP requires much more complicated schedule evaluation methods and data structures. The results for the J30 data-set are shown in Tables 4 and 6 . The CPM dev and OPT dev values are the average percentage distance from the critical path length and the average percentage distance from the optimal makespan respectively. Best sol states the number of optimal solutions which have been proved to be optimal according to the results in the PSPLIB homepage -http: //www.om-db.wi.tum.de/psplib/. Comp time is the total run-time stated in seconds and the Sched sec is the number of evaluated schedules per second. It is obvious that the GPU version is able to achieve a similar quality of solutions in terms of CPM dev as the CPU version. Having had I total doubled, the GPU version found 478 optimal solutions from the 480 solutions in the data-set. From the performance point of view Table 6 reveals a significant improvement in computational time if parallelization is performed. For example, the parallel CPU version is 3.65 times faster than the sequential CPU version and the GPU is almost 2 times faster than the parallel CPU version. If I total is increased to 20000 the GPU is still slightly faster and achieves better quality solutions. For the J60 data-set the results are shown in Tables 5 and 7 , where UB dev is the average percentage distance from the best currently known upper bounds. The CPU version gives slightly better solutions for 10000 iterations, but on the other hand if the GPU is given 20000 iterations the quality of solutions is comparable with the CPU version and the GPU is still 3.56 times faster than the parallel CPU version. The lower quality of GPU solutions for the same I total is probably caused by wasting work when many Parallel Tabu Search instances have read the same solution from the working set and only one writes the best improvement. It can be noted, that the parallel CPU version is more than 4 times faster than the sequential one. The reason of that is either better cache utilization or the AMD True Core Scalability technology.. The results in Tables 8 and 10 for the J90 dataset show that the GPU is better utilized for bigger instances and the GPU is more than 10 times faster than the parallel CPU version for the same number of iterations. The same quality of solutions is achieved 5.4 times faster on the GPU.. Results for the J120 data-set are shown in Tables 9 and 11. It can be noted that the quality of GPU solutions is substantially lower for 10000 iterations. The GPU requires about 50000 iterations to achieve the quality of the CPU solutions. On the other hand, the GPU is able to compete with the CPU since 50000 iterations is performed 2.5 times faster than 10000 iterations for the parallel CPU version. The GPU evaluates more than one million schedules per second, whereas the CPU evaluates one hundred thousand.. The quality of the solutions is compared with the existing solutions for the RCPSP in Table 12 . Our proposed PTSG outperforms other Tabu Search implementations with respect to the quality of solutions. For example, Artigues' Tabu Search [21] has been given at least 11000 iterations for the J120 data-set and achieves 36.16 % CPM dev. Having had 10000 iterations the proposed PTSG reaches 33.41 % and 34.67 % for the CPU and GPU respectively. In addition, the proposed PTSG can be just as good as other heuristic approaches like Ant Colony Optimization and Simulated Annealing. On the other hand, the state of the art random-key genetic algorithms give even better solutions than the PTSG.. From the performance point of view it is difficult to compare since the different algorithms and hardware architectures were used for experiments.. For example, Artigues's Tabu Search requires 67 s per J120 instance on average. The testing configuration was not stated. The PTSG requires 4.9 s (10000 iterations) on the mid-range GPU with the substantially higher quality of solutions. The genetic algorithm by Gonçalves et al. [18] takes 180 s per J120 instance on average on the Intel Core 2 Duo 2.4 GHz processor.

O O B I I I I O O B I I O O O B O O O O O B I I O O O B I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Some Results on Lagrangians of Hypergraphs In 1965, Motzkin and Straus [5] provided a new proof of Turan's theorem based on a continuous characterization of the clique number of a graph using the Lagrangian of a graph. This new proof aroused interests in the study of Lagrangians of r-uniform graphs. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Sidorenko and Frankl-Furedi applied Lagrangians of hypergraphs in finding Turan densities of hypergraphs. Frankl and Rodl applied it in disproving Erdos' jumping constant conjecture. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Furedi conjectured that the r-uniform graph with m edges formed by taking the first m sets in the colex ordering of $N^(r)$ has the largest Lagrangian of all r-uniform graphs with m edges. Talbot in [14] provided some evidences for Frankl and Furedi's conjecture. In this paper, we prove that the r-uniform graph with m edges formed by taking the first m sets in the colex ordering of $N^(r)$ has the largest Lagrangian of all r-uniform graphs on t vertices with m edges when ${t choose r}-3$ or ${t choose r}-4$. As an implication, we also prove that Frankl and Furedi's conjecture holds for 3-uniform graphs with ${t choose 3}-3$ or ${t choose 3}-4$ edges. In 1941, Turán [15] provided an answer to the following question: What is the maximum number of edges in a graph on n vertices without containing a complete subgraph of order t, for a given t? This is the well-known Turán theorem. Later, in another classical paper, Motzkin and Straus [5] provided a new proof of Turán's theorem based on a continuous characterization of the clique number of a graph using Lagrangians of graphs. This new proof aroused interests in the study of Lagrangians of r-graphs. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. For example, Sidorenko [11] and Frankl-Füredi [1] applied Lagrangians of hypergraphs in finding Turán densities of hypergraphs. Frankl and Rödl [2] applied it in disproving Erdös' jumping constant conjecture. More applications of Lagrangians can be found in [12] , [3] and [6] . In most applications, we need an upper bound for the Lagrangian of a hypergraph. In the course of estimating Turán densities of some hypergraphs, Frankl and Füredi [1] asked the following question: Given r ≥ 3 and m ∈ N, how large can the Lagrangian of an r-graph with m edges be? Before stating their conjecture on this problem, we give some definitions and notation.. For a set V and a positive integer r we denote by V (r) the family of all r-subsets of V . An r-uniform graph or r-graph G consists of a set V (G) of vertices and a set E(G) ⊆ V (G) (r) of edges. An edge e = {a 1 , a 2 , . . . , a r } will be simply denoted by a 1 a 2 . . . a r . An r-graph H is a subgraph of an r-graph G, denoted by H ⊆ G if V (H) ⊆ V (G) and E(H) ⊆ E(G). Let N be the set of all positive integers. For any n ∈ N, we denote the set {1, 2, 3, . . . , n} by [n] . Let K (r) t denote the complete r-graph on t vertices, that is the r-graph on t vertices containing all possible edges. A complete r-graph on t vertices is also called a clique with order t. We also let [n] (r) represent the complete r-uniform graph on the vertex set [n] . When r = 2, an r-uniform graph is a simple graph. When r ≥ 3, an r-graph is often called a hypergraph.. For an r-graph G = (V, E) we denote the (r − 1)-neighborhood of a vertex i ∈ V by E i = {A ∈ V (r−1) : A ∪ {i} ∈ E}. Similarly, we will denote the (r − 2)-neighborhood of a pair of vertices i, j ∈ V by E ij = {B ∈ V (r−2) : B ∪ {i, j} ∈ E}. We denote the complement of. Definition 1.1 For an r-uniform graph G with the vertex set [n], edge set E(G) and a vector x = (x 1 , . . . , x n ) ∈ R n , define. We call x = (x 1 , x 2 , . . . , x n ) ∈ R n a legal weighting for G if x ∈ S. A vector y ∈ S is called an. The following fact is easily implied by the definition of the Lagrangian.. In [5] , Motzkin and Straus provided the following simple expression for the Lagrangian of a 2-graph. [5] ) If G is a 2-graph in which a largest clique has order t then

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Web-based information for pregnant women and new mothers with type 1 diabetes- a description of the development process Background: This paper describes the process of developing specifically designed web-based maternity information for women with type 1 diabetes. Methods: A participatory design was used and the information was evaluated in seven stages by researchers, professional experts and users. All steps of the development process were noted in an online logbook.Results: The information developed gradually and its contents were reviewed by nurse-midwives, nurses and physicians specializing in different key areas including diabetes care, paediatrics, obstetrics and breastfeeding, a clinical dietician and mothers with type 1 diabetes. The draft was reviewed in regard to its cultural suitability and the information material was adjusted to meet quality criterions. Finally, the text was adapted for a lay audience. Conclusions: Using participatory design required time and resources, however; it proved a functional way of producing appropriate information for the target group. Development of specialized information is one aspect of providing professional support. Using participatory design proved to be a functional way of developing evidence-based information. Based on the experience gained from this project, the research team found it important to allow enough time and provide enough resources to permit the project to evolve at an unhurried pace. Getting input from different and relevant professionals not only allowed the text to evolve but also required the researchers to take time for reflection. Asking for the patients' views of accuracy and relevance was the most important step. Using participatory design helped ensure that the information on the website reflected what the women wanted to read, not just what the health professionals wanted the women to know. The comments from the women in the target group also ensured that the text appropriately reflected their transition to motherhood. More research that explores the use of participatory design in developing information is needed. There appears to be a need for validated instruments that measure cultural suitability. Developing information based on scientific evidence and on experience-based professional knowledge through a participatory design was functional, although timeconsuming. The process benefited greatly from having a heterogeneous group of experts assessing the materials independently. Having access to a group consisting of experts in medical care, popular science editing, and mothers with type 1 diabetes who had lived experience of pregnancy, childbirth and early motherhood proved indispensable because the final text would not have been as thorough and well thought out without their input.. Reaching a consensus on what to include or omit after each expert's review and revision required time for reflection on the part of the research group. Using participatory design proved an effective way of ensuring that the information provided reflected what the women wanted to read about [19] . Having input from both professionals and women/ mothers who had experience with diabetes and childbearing provided insights that were invaluable to the project team. Getting the mothers' opinions was undoubtedly the most important part of the review process, a step that could not have been left out. Their comments allowed the research team to change the content, structure and tone of the text to better suit their needs, as is vital in evidencebased information [28, 29] . Since pregnant women and women who have recently given birth are in transition to their new role as mothers [11] , it was important that the information reflect the various stages of their journey.. Developing specialized information is a challenge. There is often a gap between what the health professionals want the reader to know and what the reader is expecting to read. In traditional patient information for patients with diabetes the focus has often been on getting the patient to comply with the physician's prescribed treatment [15] . In moving towards a model of shared decision making, evidence-based information is necessary for making informed choices [28, 30] . There appears to be a lack of theory on how to measure whether patient information is comprehensible or not [17] . In this project the research team consulted a commercial company specialized in popular science editing to gain insight from professionals not biased by preunderstandings. This was helpful, but not essential. If the team had not had the insights from the target group, it probably would have missed the mark in tone and meaning. The same content may differ in perceived meaning depending on the tone of the paragraphs and headings, a valuable insight. This highlights the importance of involving members of the target group and not only skilled professionals in text development [29] .. When designing the information to suit a web setting a few considerations had to be taken into account. One is that the content must meet appropriate standards. As novices in web-based patient information development, the research team found that the Model for Online Consumer Health Information Quality [27] and Health on the Net Foundation's code of conduct [26] helped ensure that the website met an acceptable standard. The design and layout of a website also affects users' perceptions of credibility [28, 31, 32] . The recognition of well-known brands or symbols on a website help users of the site determine whether the information given is credible [32] . An overview of the seven stages is given in Figure 1 and here follows details in each stage. Pregnant women and new mothers with type 1 diabetes need both professional and personal support [1, 2] . They are at high risk for complications for both themselves and their babies, and they face a number of obstacles during the maternity period (pregnancy, labour and birth, and the first months afterwards). The increased risks include preeclampsia and complicated deliveries with foetal distress, and operational modes of delivery [3, 4] , foetal malformations, macrosomia, and neonatal complications [3, [5] [6] [7] [8] . Diabetes often overshadows the pregnancy [2, 9] due to the struggle to maintain normal blood glucose levels 24 hours a day in order to give the child the best possible chance to be born healthy. Pregnancy for women with type 1 diabetes can be dominated by feelings of worry for the child and self-blame for not giving the child the best conditions [1, 2, 9] . During labour the women worry about jeopardizing their babies' health [1] . In the early period after birth, while adapting to the maternal role and establishing breastfeeding, the women are confronted with unpredictable blood sugar levels, and many infants have hypoglycaemic episodes and other diabetes-related conditions [10] .. Becoming a mother and making the transition to motherhood is typically a time of joy but also of uncertainty, trial and error [11] . Women with type 1 diabetes are exposed to risk during pregnancy, childbirth and early motherhood, and they need extra support from their partners and relatives and from health professionals during this period [1, 9, 10] . One essential part of giving professional support is providing readily accessible information [12] . In a Swedish study conducted in 2010, of 105 recent mothers with type 1 diabetes, 12% searched the Internet every day for information concerning pregnancy, childbirth and parenthood, 29% searched one or more times a week, and 38% searched one or more times a month [13] .. Providing information through the Internet can be an effective way of supporting self-management of diabetes [14] . However, constructing patient information is not an easy task and there are a number of challenges to overcome. Traditional patient information often lacks patients' perspectives. It outlines what the experts want the patient to know and what regime they want the patients to follow [15, 16] . In contrast stands information that is designed to "empower" the reader. Even though, the idea of empowering the reader is laudable, it still reinforces the idea that the authors can control the readers' reaction to the material. Ideally, the information material should be neutral enough for the readers to make up their own minds, this is however not easy to achieve [16] . Another great challenge is to ensure that the material is comprehendible for the readers [17] .. A research program, MOtherhood and DIABetes (MODIAB), exploring and developing interventions in diabetes and childbearing [1, 10] , has identified needs of increased support in women with diabetes in the maternity period, i.e. pregnancy, childbirth and early motherhood. With this as a background a web-based support, MODIAB-Web, was designed, consisting of three main parts: 1) a self-care diary including a device for documenting and evaluating blood glucose levels, insulin doses, food intake, physical activities and overall well-being; 2) a forum for communication between women with type 1 diabetes in the childbearing period; and 3) specific maternity information in relation to type 1 diabetes. The objective of this paper is to describe the process of developing the specifically designed maternity information.

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Constraints of Second-Order Vanishing Moments on Lattice Structures for Non-separable Orthogonal Symmetric Wavelets In this paper, a design method of two-dimensional (2-D) orthogonal symmetric wavelets is proposed by using a lattice structure for multi-dimensional (M-D) linear-phase paraunitary filter banks (LPPUFB), which the authors have proposed as a previous work and then modified by Lu Gan et al. The derivation process for the constraints on the second-order vanishing moments is shown and some design examples obtained through optimization with the constraints are exemplified. In order to verify the significance of the constraints, some experimental results are shown for Lena and Barbara image. Still picture codec standards, such as JPEG and JPEG2000, and motion picture codec standards, such as MPEG-1/2 and H.264/AVC, employ transform coding techniques as basic components for reducing spatial redundancy [1] - [3] . The coding performance of whole system is severely influenced by the choice of the transform.. JPEG and MPEG-1/2 consist of a 8×8 two-dimensional (2-D) discrete cosine transform (DCT). DCT is known as a quasi-optimal transform for the first-order autoregressive process with a correlation close to 1. As well, the transform kernel is defined by the cosine function and efficient computation is available through some fast algorithms. From these reasons, DCT has favourably been used in both of still and motion picture codecs. The independent block processing, however, makes block boundaries visible and the large basis images of high frequency components causes mosquito noise for very low bit-rates. JPEG2000 tries to solve these problems by introducing discrete wavelet transforms (DWTs). The DWTs contribute not only to gain coding performances and realize the spatial scalability, but also to reduce the blocking artifacts and the mosquito noise, because of their overlapping basis images and their different sizes suitable for different frequency components.. It is well known that there is no 1-D 2-channel filter bank which simultaneously satisfies all of the overlapping, linear-phase and paranunitary (i.e. orthogonal) property [4] . It is also true for separable 2-D 2 × 2-channel filter banks, which are employed in JPEG2000. Non-separable 2 × 2-channel filter banks, however, can simultaneously satisfy all of the previous properties. As well, image and video processing requires us to pay attention to frequency support of original signals because it may be non-separable [5] . From these backgrounds, M-D non-separable systems which simultaneously satisfy the orthogonal, overlapping and linearphase property have been considered by several researches so far.. The authors also proposed non-separable transforms by using 2 × 2-channel linear-phase paraunitary filter banks (LPPUFBs) in the lattice structure as a previous work [6] . Furthermore, Lu Gan et al. showed the way of reducing the number of parameters to be optimized while preserving the class [7] . Since the orthogonal, overlapping, and linear-phase property are guaranteed, the lattice structure is convenient in terms of design and implementation. Additionally, we can construct 2 × 2-decomposition symmetric orthogonal DWTs by hierarchically applying the lattice structure. However, it has not been clarified how to design DWTs with vanishing moments through the lattice structure. Although Stanhill et al. exemplified a design procedure of such DWTs [8] , [9] , no constraint on design parameters in closed form has not been shown. For the 1-D case, the article [10] showed a design procedure of M-band orthogonal symmetric wavelets with lattice structures of second-order vanishing moments. In this work, it is shown that the constraints can be obtained in closed form of design parameters for the 2×2-channel 2-D lattice structure by using the similar approach to the 1-D lattice. Indeed, we design filter banks with our new method, and through an experiment on image coding, we verify the significance.. Organization of this paper is as follows: Section 2 reviews the design method of LPPUFBs based on lattice structure. Section 3 derives the constraints on the second-order vanishing moments for the lattice structure. Section 4 shows the results of the optimal design, followed by conclusions in Sect. 5. In this paper, a design method of 2-D orthogonal symmetric wavelets was proposed by using a lattice structure for M-D LPPUFB, which the authors have proposed before and Lu Gan et al. have modified later. The derivation process of the constraints on the second-order vanishing moments was shown and some design examples obtained through optimization with the constraints were shown. By using the designed filter banks, we verified the significance through experimental results of image coding. In future, we will consider increasing the order of vanishing moments and investigate the filter directionality.

O O O B I O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O