Time-stamped resampling for robust evolutionary portfolio optimization Traditional mean-variance financial portfolio optimization is based on two sets of parameters, estimates for the asset returns and the variance-covariance matrix. The allocations resulting from both traditional methods and heuristics are very dependent on these values. Given the unreliability of these forecasts, the expected risk and return for the portfolios in the efficient frontier often differ from the expected ones. In this work we present a resampling method based on time-stamping to control the problem. The approach, which is compatible with different evolutionary multiobjective algorithms, is tested with four different alternatives. We also introduce new metrics to assess the reliability of forecast efficient frontiers. Asset allocation is one of the core topics in financial management. The search for the optimal choice of financial assets to be included in a portfolio has been the subject of research for a long time and it is one of the most active lines in finance.. The academic literature on this subject is very large and mostly based on the work of Markowitz (1952) and Markowitz (1959) . Under this framework, the problem is introduced as a multiobjective optimization problem where the investor tries to figure out the weight that each of the investment alternatives should carry in the portfolio. The target of this investor would be both minimizing risk and maximizing return at the same time. The solution to the problem of optimizing for these two objectives in conflict defines a set of solutions called the efficient frontier. This Pareto front consists on portfolios that are neither better or worse than the rest. For each level of risk or return, there is no better alternative in terms of the other objective. This makes the election of one of them a choice to be made by investors according to the way they weight risk and rewards.. The basic version of the problem can be solved using Quadratic Programming (QP). Unfortunately, this approach is built on a set of assumptions that are unlikely to hold in the real world. The quest for alternatives has driven attention to metaheuristics that might not suffer this limitation. This is the reason why the framework of evolutionary computation is getting traction on this area.. The solutions based on metaheuristics mostly fall into one this two categories: solutions that transform the multiobjective problem into a single-objective form, and those who deal with it using a mutiobjective approach.. Among the first group, we would mention the work by Soleimani, Golmakani, and Salimi (2009). These authors use a genetic algorithm and extend the mentioned classic mean-variance optimization model and consider constraints on transaction costs, round lots or cardinality. They tackle the multiobjective nature of the problem minimizing the risk objective while setting a range of minimum acceptable returns in the constraints. Chiranjeevi and Sastry (2007) use another popular approach to transform the multiobjective problem into a version that can be handled by single-objective algorithms. Instead of keeping one of the objectives on the objective function and the other in the constraint, they consider the objectives in the fitness function weighting them. In this particular case, they manage five objectives that are obtained breaking down the basic two. Chang, Yang, and Chang (2009) suggest a similar solution and optimize a function that weights risk and return using a risk aversion parameter. Zhu, Wang, Wang, and Chen (2011) target a popular metric, the Sharpe Ratio, that combines both elements into a single expression.. The development of multiobjective evolutionary algorithms (MOEAs) has resulted into a number researchers exploring their performance in this area. Among these we could mention Skolpadungket, Dahal, and Harnpornchai (2007) , who test a set of multi-objective algorithms (VEGA, SPEA2, NSGA-II etc.) on a constrained version of the two objective problem. More recently, Anagnostopoulos and Mamanis (2011) compare the performance of different multiobjective algorithms, and Deb, Steuer, Tewari, and Tewari (2011) introduce a customized hybrid version of NSGA-II to tackle the problem. Finally, we will mention the work of Radziukyniene and Xilinskas (2008), where authors compare FastPGA, MOCELL, AbYSS, and NSGA-II on both of the basic problems, and an extended version that considers the dividend yield as a third objective.. Despite of the amount of research on portfolio optimization, there are still open issues. A key one is the robustness of results provided by algorithms. Among the most important factors that asset managers have to consider when evaluating the results provided by any of the above-mentioned methods, there is reliability. Very often, the expected efficient frontier lies far from the real one. This is due to the fact that the estimates for the expected risk and returns for the assets in the solution, and the portfolios derived from them, are very inaccurate. Understandably, this results on mistrust by some practitioners. The search for solutions for this problem has cleared the way for the field of robust portfolio optimization. It is in this area where we focus our contribution. We introduce a time-stamping method to control the population that enhances the reliability of the solutions provided by MOEAs.. The process of optimizing the risk and return of a portfolio relies on two parameters: the estimates for the expected asset returns and the variance-covariance matrix. The values of these parameters are usually based on past data and they might be inaccurate due to, for instance, the presence of outliers. In this context, there are several potential ways to tackle the problem. The two most prevalent alternatives found in the literature rely on either putting an emphasis on having robust estimates for the parameters, or managing the optimization process itself. The first one usually tries to filter the estimates to control, for instance, the influence of extreme past events on their computation (Perret-Gentil et al., 2005) . The authors focusing on the second alternative design approaches handle uncertainty in the parameters during the optimization process (Pflug & Wozabal, 2007; Tütüncü & Koenig, 2004) . The alternative suggested in this paper falls in the latter category. We will enhance the solutions of MOEAs testing the population for different values for the parameters, and selecting the portfolios that consistently offer a good performance.. Optimizing for a single scenario, a set of expected asset returns and the use of a single variance-covariance matrix, bears the risk of getting solutions that might be extremely sensitive to deviations. This could potentially be a problem as it is almost certain that the estimates will not be accurate. We have to bear in mind that having perfect estimates for the expected returns for instance, implies that we can make perfect predictions for future prices, which is highly unlikely. For this reason we consider that assessing the candidate solutions in different likely scenarios and favoring those that consistently offer a good performance, might be a promising approach. The requirement of consistency is key. In order to achieve it, we introduce a time-stamping mechanism that, together with performance in terms of risk and return, will drive the evolution process to find robust and stable solutions.. The approach introduced in this paper is related to alternatives based on resampling (Ruppert, 2006; Shiraishi, 2008) . The most comparable among the traditional approaches is the one described by Idzorek (2006) . This author suggests using combining traditional QP with Monte Carlo simulation to derive a set of fronts that are merged into a single solution at a later stage. This idea is very interesting but, unfortunately, the approach suffers the shortcomings of QP, namely, the ability to deal with real-world constraints. This is the reason why we feel that adapting the idea to the framework of MOEAs is very promising, as they do not suffer from this limitation. There is a previous effort based of evolutionary algorithms along the lines of this work, but is based on a very simple resampling approach that optimizes for a different scenario in each generation (Garcia, Quintana, Galvan, & Isasi, 2011) . The time-stamping mechanism that we introduce in this work is based on the previous one extending the problem with a third implicit objective that favors solutions that are consistently reliable. As we will see in the experimental section, this results on significantly higher robustness.. The proposed approach is compatible with a wide array of MOEAs. Given their different nature and behavior, we will test the approach on a set of popular algorithms. Specifically, the experimental section will consider NSGA-II, SPEA2, SMPSO and GDE3. NSGA-II (Deb, Pratap, Agarwal, & Meyarivan, 2002) is one the most referenced algorithms in the field of multiobjective optimization. This one, together with SPEA2 (Zitzler, Laumanns, & Thiele, 2001) have been confirmed (Skolpadungket et al., 2007) to offer good performance in portfolio optimization. Apart from the mentioned two, we consider GDE3 (Kukkonen & Lampinen, 2005) , a differential evolution strategy, and SMPSO, (Nebro et al., 2009 ) a multiobjective algorithm based on particle swarm optimization.. In the context of multiobjective problems, an important issue is the metric used to evaluate the solutions. It is generally admitted that there is no ideal single metric that should be used to evaluate different objectives simultaneously in every circumstance. The metrics that are most commonly used in this field, such as Hypervolume, Spread or SetCoverage are not appropriate indicators of stability in this context. For this reason, we define a set of metrics that capture different aspects of robustness in efficient frontiers. These, Estimation Error, Stability, Extreme Risk and Unrealized Returns draw on the basic principle that the expected risk and returns for the portfolios in the solution should be close to the observed ones ex-post.. The rest of the paper is organized as follows. First, we make a formal introduction to the financial portfolio optimization problem. After, we describe in detail the evolutionary approach proposed in this work. This section includes a brief description of the MOEAs, the solution encoding and the fitness mechanism in order to find robust and stable portfolios. Next, the different metrics used in this work to evaluate robustness of solutions are described. That will be followed by the experimental results and, finally, there will be a section devoted to summary and conclusions.

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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


Steady and fair rate allocation for rechargeable sensors in perpetual sensor networks Renewable energy enables sensor networks with the capability to recharge and provide perpetual data services. Due to low recharging rates and the dynamics of renewable energy such as solar and wind power, providing services without interruptions caused by battery runouts is non-trivial. Most environment monitoring applications require data collection from all nodes at a steady rate. The objective of this paper is to design a solution for fair and high throughput data extraction from all nodes in presence of renewable energy sources. Specifically, we seek to compute the lexicographically maximum data collection rate for each node, such that no node will ever run out of energy. We propose a centralized algorithm and an asynchronous distributed algorithm that can compute the optimal lexicographic rate assignment for all nodes. The centralized algorithm jointly computes the optimal data collection rate for all nodes along with the flows on each link, while the distributed algorithm computes the optimal rate when the routes are pre-determined. We prove the optimality for both the centralized and the distributed algorithms, and use a testbed with 155 sensor nodes to validate the distributed algorithm. Recent years have witnessed a growing number of prototype sensor network deployments for environment monitoring such as earthquake monitoring [1] , structural monitoring [2, 3] , soil monitoring [4] , and glacial movement monitoring [5] . Unattended operability for long periods is highly desirable due to the harshness of the environment, which is typical in such applications. However, the limitations of fixed battery sources often require operation at low data rates, which may even be unacceptable to some applications.. Energy harvesting from natural sources such as solar [6, 7] , wind [8] , thermal [9] , and vibration [10, 11] , have been shown to be effective in addressing the above problem. The amount and source of energy that can be leveraged depends on the environment and the application. For example, in a deployment for monitoring vibration of industrial machines, vibration energy could be readily available. Similarly, for a deployment for monitoring the air-quality in the AC-ducts, thermal energy can be harvested when hot air is blowing. For outdoor deployments, solar energy is readily available for a wide range of application scenarios.. Environment monitoring applications often require periodic collection of data from all nodes. The objective of this paper is to design a solution for fair and high throughput data extraction from all the nodes in the network in presence of renewable energy sources. Specifically, we seek to compute the lexicographically maximum data collection rate for each node in the network, such that no node will ever run out of energy. A rate assignment is lexicographically maximum if it is impossible to increase the rate of a node without decreasing the rate of another one with a lower rate.. The time varying profile of recharging rates poses challenges in the computation of the optimum data rates. If the rate of data collection is high, a node can end up depleting its battery, which can hurt coverage as well as network connectivity. If the network gets partitioned due to a node running out of energy, the amount of data collected during that period also gets impacted. Similarly, if the data collection rate is low, the battery may reach the highest level which results in missed recharging opportunities. The determination of optimum data rates must take into account the profile of recharging, together with the energy cost of sensing, transmission, and packet reception.. The contributions of the paper are as follows:. • We propose a centralized algorithm to compute the optimum data collection rate for each node, along with the amount of flow on each link in the network.. • We propose an optimum distributed and asynchronous algorithm assuming that the routing tree is pre-determined.. • We prove the optimality of the centralized and the distributed algorithms.. • Using experimentation with solar panels and a sensor network testbed, we evaluate the performance of the proposed approaches under various realistic scenarios.. The organization of the paper is as follows. Section 2 presents the centralized algorithm to compute the optimal lexicographic rate assignment and corresponding routing paths. Section 3 presents the distributed algorithm. The evaluation of the protocols using experiments with solar energy harvesting and experiments on a testbed are presented in Section 4. We discuss some issues not addressed in this paper and our future works in Section 5. Section 6 presents background and related work. Finally, Section 7 concludes the paper. In this paper we study the rate assignment problem for rechargeable sensor networks. We propose a centralized algorithm and a distributed algorithm for optimal lexicographic rate assignment. The centralized algorithm computes the optimal rate assignment along with determining the amount of flow on each link. The distributed algorithm computes the optimal rate when the routing tree is predetermined. We prove the optimality of the centralized and the distributed algorithms. To evaluate the proposed distributed algorithm, we conduct experiments using a testbed with 155 sensor nodes under various scenarios. The rates computed by the distributed algorithm are dependent on the routing tree. As part of our future work, we plan to explore distributed solutions that can jointly determine the optimum rates for all nodes and the flows on each link.. First, we define some terms that will be used in the proof. Definition 2. Let G(V, E, S) be a network with nodes V = {v1, v2, ..., vn}, wireless links E, and node constraints S = {s1, s2, ..., sn}. A node constraint si is a 2-tuple of (Πi, Ri) that specifies the maximum battery capacity and energy recharging rate for vi. We define G (N , E , S ) = G 2 (N, E, S) where N = N , E = E, S = {s 1 , s 2 , ..., s n }, and s i = 2 × si = (2Πi, 2Ri), 1 ≤ i ≤ n. In short, G 2 is a network with the same topology as G, but the maximum capacity and energy recharging rate of each node is doubled in G 2 . We also define G = G 2 /2.. Definition 3. Let L1 and L2 be any two rate assignments for nodes {v1, v2, ..., vn} where L1 = {r n }. We define L = L1 + L2 where L = {r 1 , r 2 , ..., r n } and r i = r. Lemma 1. The lexicographic optimal rate assignment is unique.. Proof. We prove the uniqueness by contradiction. We show that if there are more than one optimal solutions, say L1 and L2, there must exist a rate assignment that is better than L1 and L2 so they can not be optimal.. Suppose there are two different optimal rate assignments, L1 and L2, for a network G. L1 and L2 are identical as sorted vectors but their rate assignments for some nodes differ. Let M be the set of nodes whose rates are different in L1 and L2. Among all the nodes in M , let v k be a node with the smallest rate in L1. Let S1 = {vi | r. k }, i.e., S1 contains all nodes whose rates are smaller than v k 's rate in L1, Q1 contains all nodes whose rates are equal to v k 's rate in L1, and X1 contains all nodes whose rates are larger than v k 's rate in L1. Next, we define S2 = {vi | r (2) i < r (1) k }, i.e. S2 contains all nodes whose rates in L2 are smaller than v k 's rate in L1. We know that. because v k is the node with the smallest rate in L1 that differs in its rate assignment in L2. Therefore,. Furthermore,. otherwise L1 and L2 will have different number of nodes whose rates are smaller than r. k , which contradicts that L1 = L2. By (14) and (15), we have. Now we construct a new network G 2 . Observe that. We show that L is lexicographically greater than L1, and therefore L1 can not be optimal.. First, consider the rates in L for nodes in S1. By Definition 3 and (13) we have. Considering the rates in L for nodes in Q1 and X1. First. otherwise vi would be in S2, which indicates vi ∈ S1 and is a contradiction that vi ∈ Q1 ∪ X1. Therefore, by Definition 3 and (18) we have. k }, and X = {vi | r i > r (1) k }. By (17) , (19) , and (20) we know S1 ⊆ S (21). Because S1 ∪ Q1 ∪ X1 = S ∪ Q ∪ X , by (21) and (22) Q1 ⊇ Q. By (19) and (20). Because S ∩ (Q ∪ X ) = φ and S1 ∩ (Q1 ∪ X1) = φ, we have. From (21) and (24) we know that |S1| ≤ |S | and |Q1 ∪ X1| ≤ |Q ∪ X |. By (25), we know |S1| = |S | and this together with (21) gives us. Furthermore, r. k > r. (1) k otherwise v k would be in S2 which indicates v k ∈ S1, a contradiction of the definition of S1. Therefore, r k = (r. k . Since v k ∈ Q1 and v k ∈ X , using (23) we further have. By (26), (27) , and (28), L > L1, which is a contradiction that L1 is an optimal lexicographic rate assignment. There are many works on developing sensors with capability of harvesting energy from solar or wind resources [8, [18] [19] [20] . There are also many studies on exploiting renewable energy to increase system performance or network lifetime [21] [22] [23] [24] [25] [26] [27] [28] . In [22] , the authors consider solar-aware routing in rechargeable sensor networks. They use a simple heuristic that preferably routes packets through solarpowered nodes, and the extra harvested energy is only a means to boost the network lifetime. In [24] , the authors measure the environmental energy properties and renewable opportunities, and use the information to schedule tasks to increase the network lifetime. In [25] and [26] , the authors further consider maximizing the system performance while maintaining Energy-Neutral operation, i.e., the energy used is always less than the energy harvested so that the system can continue to operate perennially. In [27] , the authors study how sensor nodes should be activated dynamically so as to maximize a utility function based on the coverage area of the sensors. In [28] , in addition to adjusting the duty cycle of the sensors to achieve Energy-Neutral operation, the authors also consider the variability of environmental energy resource and attempt to reduce the variation of the duty cycle using adaptive control theory. However, these works either only consider the workload of individual sensors and do not consider the influence on overall network performance by the individual decisions, or only try to maximize the system performance but do not consider the impact on individual sensors.. To maximize the system performance while balancing the workload, fairness has to be considered. Maxmin fairness, or lexicographic fairness, has been widely used to define the fairness of a system. In [14] , a distributed maxmin rate computation algorithm for fixed flows routed through capacity constrained switches in wired networks is proposed. The proposed algorithm computes a maxmin rate assignment for each flow and guarantees quick convergence. In [29] , the authors generalize the problem by adding maximum and minimum rate requirement for each flow, and propose a centralized algorithm similar to the one proposed in [30] that identifies bottleneck links first, and assigns rates equally to all flows passing through these bottleneck links. A distributed algorithm is also proposed that is based on the algorithm proposed in [14] . In [13] , a centralized algorithm is proposed that iteratively uses linear programming to find lexicographic rate assignment for all sensor nodes that periodically report readings to the sink. The proposed algorithm does not require these flows to be forwarded through fixed routes. In [31] a distributed congestion control scheme is proposed to achieve maxmin rate allocation through overhearing and propagating congestion announcement, but it requires sophisticated parameter tuning to achieve stable operation and the rates oscillate up and down after converging even when the topology does not change. All these works solve the fairness problem with static constraints, such as based on switches or battery capacity, and do not consider dynamic resources such as changing harvested energy.

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


Generalized Silver Codes Abstract-For an n t transmit, n r receive antenna system (n t 2 n r system), a full-rate space time block code (STBC) transmits at least n min = min(n t ; n r ) complex symbols per channel use. The well-known Golden code is an example of a full-rate, full-diversity STBC for two transmit antennas. Its ML-decoding complexity is of the order of M 2:5 for square M -QAM. The Silver code for two transmit antennas has all the desirable properties of the Golden code except its coding gain, but offers lower ML-decoding complexity of the order of M 2 . Importantly, the slight loss in coding gain is negligible compared to the advantage it offers in terms of lowering the ML-decoding complexity. For higher number of transmit antennas, the best known codes are the Perfect codes, which are full-rate, full-diversity, information lossless codes (for nr nt) but have a high ML-decoding complexity of the order of M n n (for n r < n t , the punctured Perfect codes are considered). In this paper, a scheme to obtain full-rate STBCs for 2 a transmit antennas and any n r with reduced ML-decoding complexity of the order of M n (n 0 )00:5 is presented. The codes constructed are also information lossless for nr nt, like the Perfect codes, and allow higher mutual information than the comparable punctured Perfect codes for nr < nt. These codes are referred to as the generalized Silver codes, since they enjoy the same desirable properties as the comparable Perfect codes (except possibly the coding gain) with lower ML-decoding complexity, analogous to the Silver code and the Golden code for two transmit antennas. Simulation results of the symbol error rates for four and eight transmit antennas show that the generalized Silver codes match the punctured Perfect codes in error performance while offering lower ML-decoding complexity.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Deterministic approximate inference techniques for conditionally Gaussian state space models We describe a novel deterministic approximate inference technique for conditionally Gaussian state space models, i.e. state space models where the latent state consists of both multinomial and Gaussian distributed variables. The method can be interpreted as a smoothing pass and iteration scheme symmetric to an assumed density filter. It improves upon previously proposed smoothing passes by not making more approximations than implied by the projection onto the chosen parametric form, the assumed density. Experimental results show that the novel scheme outperforms these alternative deterministic smoothing passes. Comparisons with sampling methods suggest that the performance does not degrade with longer sequences. Many real-world problems can be described by models that extend the classical linear Gaussian dynamical system with (unobserved) discrete regime indicators. In such extended models the discrete indicators dictate what transition and observation model the process follows at a particular time point. The problems of tracking and estimation in models O. Zoeter ( ) · T. Heskes Biophysics, Radboud University Nijmegen, Geert Grooteplein 21, 6525 EZ, Nijmegen, The Netherlands e-mail: o.zoeter@science.ru.nl T. Heskes e-mail: t.heskes@science.ru.nl with maneuvering targets (Bar-Shalom and Li, 1993) , multiple targets (Shumway and Stoffer, 1991) , non-Gaussian disturbances (Kitagawa, 1996) , unknown model parameters (Harrison and Stevens, 1976) , failing sensors (Lerner et al., 2000) and different trends (Hamilton, 1989) are all examples of problems that have been formulated as conditionally Gaussian state space models. Since the extented model is so general it has been invented and re-invented many times in multiple fields.. Although the extended model is very powerful, it is notorious for the fact that exact estimation of posteriors is intractable. In general, exact filtered, smoothed or predicted posteriors have a complexity exponential in the number of observations (Lerner and Parr, 2001) . Section 2 describes the intractability in more detail.. In this article we introduce a deterministic approximation scheme that is particularly suited to find smoothed one and two-slice posteriors. It can be seen as a symmetric backward pass and iteration scheme for previously proposed assumed density filtering approaches (Harrison and Stevens, 1976) .. The intractability problems are shared between all variants of the conditionally Gaussian state space model. In Section 2 we introduce the general model; variants where only the transition or only the observation model switch, or where states or observations are multi-or univariate can be treated as special cases. The assumed density filtering approach that forms the basis for our approximation scheme is described in Section 3. In Section 4 the symmetric backward pass is introduced. Since both the forward and the backward pass consist of local, greedy projections it makes sense to iterate them. Section 5 introduces the objective that is minimized by such an iteration scheme and gives an intuition how we should interpret fixed points. In Section 6 we describe two approximate smoothing passes that are often used for conditionally Gaussian state space models. In Section 7 related deterministic approximation are discussed. Section 8 describes experiments that test the validity of the proposed method and compare it with the alternative backward pass and state-of-the-art sampling techniques.

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


Efficient compression and handling of current source model library waveforms Abstract-This paper describes a waveform compression The traditional, delay and slew based, library cell modeling methodology [1] is no longer accepted as accurate for the new nanometer-era CMOS technologies. Recently, the industry is in the process of adopting the more detailed Current Source Modeling (CSM) as an alternative. ECSM [2] (initially developed by Cadence Design Systems and adopted by SI2 Open Modeling Coalition), and CCS [3] (proposed by Synopsys) are extensions of the Liberty library format and contain separate models for timing, noise, and power applications.. The new CSM modeling paradigm combined with technology trends require significantly more memory resources compared to traditional techniques. CSM modeling requires the storage of tables of current or voltage waveforms in addition to just the delay and slew quantities. Additionally, the increased parameter variability inherent in modern VLSI technologies, forces the timing analysis to rely on library characterizations even more points in the temperature, voltage and process space, thus further compounding the library size problem. Finally, the noise and power verification tools are also in the process of converting to waveform-based models, causing a further explosion of the library modeling data.. Library characterization tools already attempt waveform compression in various ways, e.g., Nangate public library stores waveforms with a minimal number of points, but on a non-uniform voltage grid [4] . This effectively doubles the information that needs to be stored. Synopsys Liberty NCX uses an empirical waveform compression scheme that takes into account key features of the current vs. time waveforms [1] . As such, their method is very specific to their particular modeling methodology. Authors in [5] represent a compact variational model waveform to be used during statistical static timing analysis, by storing the nominal waveform and presenting any perturbed waveform using time shifting, time scaling, voltage shifting, and voltage scaling operations. However, this paper does not store the nominal waveform, itself, in a compact form, which requires large memory resource during timing analysis. The authors in [6] model the voltage waveforms of the gates as a linear combination of a fixed set of basis waveforms chosen by a singular value decomposition (SVD) algorithm. However, as it is discussed in the present paper, this approach might result in non-causal waveforms for some complex gates in the CMOS library which contains transmissiongates. Thus, in our paper, we address adaptive compaction of current-source model libraries by representing each waveform using a variable number of basis waveforms, i.e. using 2-3 basis waveforms to represent a large set of waveforms and using up to 14 basis waveforms for a very small set of waveforms. Adaptive compression also results in higher compression ratios than a fixed set representation.. The compression introduced in this paper is based on rigorous and general theory; it is by certain measures optimal, and can be extended to all library formats. Our method consists of constructing a basis of orthogonal waveforms from the largest available collection of the waveforms stored in the library. Adaptive compression is achieved by representing each waveform in the library as a linear combination of a variable number of basis waveforms. Therefore only a few coefficients need to be stored for each waveform from which it can be reconstructed with great accuracy. The generation of the basis consists of a carefully tuned procedure consisting in shifting, scaling, averaging, weighting, and performing SVD on the entire collection of waveform data. The procedure is accurate and efficient and can accommodate large waveform collections, as the computational complexity increases almost linearly with the number of waveforms. Note that the causality of the compressed waveforms is guaranteed during the compression procedures.. In addition, the coefficients chosen to represent the waveform exhibit a smooth dependence on variables of interest such as load capacitance, input slew, temperature, etc., and therefore can be parametrized using simple analytic functions. Besides significantly increasing the compression ratio, the other important benefit of parameterization is to provide the necessary interpolated CSM waveforms at points not stored in the model tables as needed by timing algorithms [7] .. We demonstrate the effectiveness of the proposed method by compressing different real, industrial, CSM libraries. The achieved compression is from 75% to over 95% depending on the required accuracy. A linear-algebra-based compression technique, PCA, was utilized to reduce the size of the CSM libraries. The technique was applied to several 65nm, ECSM and CCS libraries. The simulation results proved the high accuracy and large compression ratio of the proposed technique. Moreover this paper presented a parametriccompression which results in further compression and an analytical interpolation model for waveform computation. In future we plan to augment the PCA-based parameterized model considering voltage, temperature, and process-related sources of variations. Library cells are pre-characterized in the following way. Circuit-level simulations are performed on the CMOS gates, excited with voltage ramps with a range of slew values and loaded by a range of pure capacitance values. CSM library standards differ by the forms in which they store the simulation results. ECSM based characterization uses tables of cell voltage response tables. The characterization data is stored as tables of time-voltage waveforms (represented by time delays at which cell outputs cross a set of pre-defined voltage thresholds) for each combination of input slew and load capacitance, as shown in Figure 1 . (a).. CCS uses a similar characterization style; the main difference is that the characterization data is stored as tables of current (rather than voltage) as a function of time, as depicted in Figure 1 . (b). The two raw characterization data sets are essentially equivalent and can be mapped from one to another. Table. The storage of entire waveforms in the CSM methodology as opposed to just selected features (e.g., delay and slew) in traditional techniques represents an order-of-magnitude increase of the characterization data volume and calls for data compression, even at the cost of some accuracy. We adopt a common, and wellknown, method for data compression based on Principal Component Analysis (PCA). The essence of PCA is the representation of the given data in a new coordinate system using a linear transformation. In the new system, the subspace of highinformation-content data is easily distinguished from the subspace of low-information-content or redundant data.. PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by any projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. PCA can be used for dimensionality reduction in a data set by retaining those characteristics of the data that contribute most to its variance. Compression is achieved by keeping lower-order principal components and ignoring higher-order ones. The loworder components contain the "most important" aspects of the data. PCA is theoretically the optimum transform for fitting a given data set in the least squares sense (L 2 norm). It is also efficiently computed using the SVD algorithm [8] .

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 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 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 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


Optimal Selective Feedback Policies for Opportunistic Beamforming This paper studies the structure of downlink sum-rate maximizing selective decentralized feedback policies for opportunistic beamforming under finite feedback constraints on the average number of mobile users feeding back. First, it is shown that any sum-rate maximizing selective decentralized feedback policy must be a threshold feedback policy. This result holds for all fading channel models with continuous distribution functions. Second, the resulting optimum threshold selection problem is analyzed in detail. This is a nonconvex optimization problem over finite-dimensional Euclidean spaces. By utilizing the theory of majorization, an underlying Schur-concave structure in the sum-rate function is identified, and the sufficient conditions for the optimality of homogenous threshold feedback policies are obtained. Applications of these results are illustrated for well-known fading channel models such as Rayleigh, Nakagami, and Rician fading channels. Rather surprisingly, it is shown that using the same threshold value at all mobile users is not always a rate-wise optimal feedback strategy, even for a network in which mobile users experience statistically the same channel conditions. For the Rayleigh fading channel model, on the other hand, homogenous threshold feedback policies are proven to be rate-wise optimal if multiple orthonormal data carrying beams are used to communicate with multiple mobile users simultaneously.Index Terms-Majorization, opportunistic beamforming (OBF), selective feedback, sum-rate, vector broadcast channels. In this paper, we have focused on the structure of optimal selective decentralized feedback policies for OBF under finite feedback constraints on the average number of MUs feeding back. First, we have shown that threshold feedback policies in which MUs compare their beam with a threshold for their feedback decisions are always optimal to maximize the downlink sum-rate. This result provides a formal justification for the use of homogenous threshold feedback policies in previous works, and holds for all fading channel models with continuous distribution functions.. Having established the optimality of threshold feedback policies, we now face an optimal threshold selection problem to maximize the sum-rate. This is a nonconvex optimization problem over finite dimensional Euclidean spaces. We have solved this problem by identifying an underlying Schur-concave structure in the sum-rate. Specifically, we have obtained sufficient conditions ensuring the Schur-concavity of the sum-rate, and therefore the rate optimality of homogenous threshold feedback policies. These sufficient conditions have been provided for general fading channel models as well.. Finally, we have performed an extensive numerical and simulation study to illustrate the applications of our results to familiar fading channel models. With some surprise, we have shown that homogenous threshold feedback policies are not always optimal to use, even when all MUs experience statistically the same channel conditions. In the particular case of Rayleigh fading channels, on the other hand, homogenous threshold feedback policies have been proven to be rate-wise optimal if multiple beams are used for the downlink communication. We have also studied the optimality and suboptimality regions for the homogenous threshold feedback policies in the Rician and Nakagami case. The detailed insights regarding when and why homogenous threshold feedback policies are rate-wise optimal or suboptimal have been provided, in conjunction with various other design and engineering perspectives. , both feedback policies will achieve the same throughput by scheduling the MU having . Therefore, we must have on the loss event. Now, if , MU 1 will not feed back under , which implies no potential loss on beam 1. Therefore, for all , we must have and . If and , MU 1 requests beam 1 under both feedback policies, resulting in a neutral event. This implies that and for all . Therefore, we also have , which concludes the proof. This section briefly reviews the papers which are most relevant to ours. In this paper, we are motivated by opportunistic communication and random beamforming techniques and focus on the downlink sum-rate maximization under finite feedback constraints. For multibeam OBF, the capacity scaling laws were first obtained by Sharif and Hassibi in [2] . They, in particular, showed that if an opportunistic scheduling algorithm is used to harvest multiuser diversity gains, the downlink throughput scales optimally like , where is the number of transmit antennas at the BS, and is the number of MUs in the system. In [3] , tighter expressions for the downlink sum-rate scaling for OBF were derived.. However, without any user selection, the number of MUs feeding back grows linearly with the total number of MUs in the system to achieve the above double logarithmic growth in the downlink sum-rate. Threshold feedback policies are frequently used to alleviate such an excessive feedback requirement [4]- [7] , [22] , [23] . In [4] , the authors used a constant threshold level, the same for all MUs and independent of the number of MUs, to reduce the total feedback load for vector broadcast channels within the OBF framework. Such a constant thresholding scheme cannot eliminate the linear growth in the average number of MUs feeding back. In [5] , it was shown that it is enough to have only MUs feeding back to achieve the same downlink sum-rate scaling in [2] by varying the common threshold level with the total number of MUs in the system. The threshold was set such that any user having an above the threshold will achieve scaling, leading to the notion of one bit of feedback per MU.. Recently, this one bit feedback result was extended in [6] by showing that , , MUs are enough to achieve the same downlink sum-rate scaling in [2] . It is almost as if constant feedback load is enough to maintain optimum sum-rate scaling but not exactly. Furthermore, if the asymptotical optimality is relaxed to -optimality in the fixed regime, selecting only MUs through thresholds is enough to achieve -portion of the full CSI sum-rate capacity [7] . It was also shown in [7] that OBF is asymptotically feedback optimal in the sense that it achieves -portion of the full CSI sum-rate capacity with minimum feedback requirement in the fixed regime. The fixed regime is the regime of interest in our setup, and there is no such a feedback optimality result for any other transmit beamforming strategy in this regime. Since one of our main goals in this paper is to understand the achievable fundamental capacity limits in vector broadcast channels subject to finite feedback constraints, a transmit beamforming strategy other than OBF would possibly be suboptimal to start with and would lead to inaccurate sum-rate estimates in our setup when there are large numbers of MUs in the network. In [22] and [23] , the amount of feedback reduction by the use of threshold feedback policies was studied for single antenna systems. This work differs from the above existing work on several important fronts. To start with, the results derived for the structure and optimization of the downlink sum-rate in this paper are correct for any number (small and large) of MUs in the network. We provide an analytical justification for why threshold feedback policies are the right choice for user selection in finite networks, e.g., see Section IV for details. Second, we pose and solve an optimum threshold selection problem in which we search for the optimum assignment of thresholds to MUs under finite feedback constraints (i.e., contrast this with the growing feedback requirement in above papers). We show that using the same threshold value for all MUs is not always optimum even if all MUs experience statistically the same channel conditions. Finally, most of our results are provided for general statistical models of vector broadcast channels, not only for the Rayleigh fading case.. An important issue associated with OBF is its applicability to finite networks. The previous works [8] - [10] proposed various methods for optimizing OBF for smaller sets of MUs. In [8] , Kountouris et al. put forward algorithms to select a target group of MUs and then to request perfect CSI only from the selected set of MUs to facilitate more efficient beamforming schemes. In [9] and [10] , it was shown how feedback aggregation and multiple beamforming vectors can be utilized to fine-tune OBF, respectively. In [11] , CSI parameters were quantized to reduce the feedback load for OBF, which cannot eliminate the linear feedback load growth alone but leads to further feedback reductions when combined with a user selection protocol. In this paper, we solve the optimum threshold selection problem offline under statistical channel information. Once the thresholds are optimally assigned, quantizing the parameters is an added design choice, and the resulting performance analysis requires further investigation, which we do not address in this paper.. Thresholding techniques are also commonly employed in other works that do not focus on OBF such as [20] and [21] . In [20] , the authors studied zero-forcing beamforming and used an orthogonality threshold for selecting MUs. Optimum sum-rate is achieved asymptotically. In [21] , again by using zero-forcing beamforming, only a subset of MUs having channel vectors nearly orthogonal to each other and singular values above a given threshold value were selected for communication. Necessary and sufficient conditions to achieve optimum sum-rate capacity with MU selection were provided.. Fairness is also among the important topics for OBF. Proportionally fair algorithm proposed in [1] ensures long-term fairness among MUs in terms of average data rates achieved. Although indirectly, this paper reveals an interesting and somewhat counterintuitive observation in regards to fairness in the OBF framework. Even for a network with statistically identical MUs, we show that it may become more favorable to treat MUs unequally, i.e., to prefer one group of MUs over others by allocating the wireless channel to them more frequently, to maximize the downlink sum-rate. We obtain various sufficient conditions on wireless channel statistics under which fairness is automatically achieved, i.e., all MUs are given equal chances to feed back and to access the channel, while maximizing the downlink sum-rate.

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


Identifying table boundaries in digital documents via sparse line detection Most prior work on information extraction has focused on extracting information from text in digital documents. However, often, the most important information being reported in an article is presented in tabular form in a digital document. If the data reported in tables can be extracted and stored in a database, the data can be queried and joined with other data using database management systems. In order to prepare the data source for as a specific document component, is widely used in web pages, scientific documents, financial reports, etc. Researchers always use tables to concisely display the latest experimental results or statistical financial data in a condensed fashion. Other researchers, for example, who are conducting the empirical studies in the same topic, can quickly obtain valuable insights via examining these tables. Along with the rapid expansion of the Internet, tables become a valuable information source in the information retrieval field. Based on the increasing demands to unlock the information in tables, more applications appear in the table-related fields, e.g., the table search [12] . Although approaches on table analysis are diverse, they share two same analyzing steps: table boundary detection and table structure decomposition. For the further table content storage and sharing (e.g., the table data extraction and the table search), locating the table boundary in a document is the first and crucial step.. Different from most table detection works, which are the pre-defined layout based and the rule-based methods, we apply machine learning techniques on the table boundary detection field in this paper. Pre-defined layout based algorithms usually work well for one domain, but are difficult to extend. For the rule-based methods, the performance is always heavily affected by the quality of the rules. When the testing data set is large enough, it is difficult to determine the "good" values for thresholds. Machine learning methods are good choices to deal with such problems. Wang et al. [23] applied the decision tree and Support Vector Machine (SVM) techniques to classify the web tables into genuine tables and non-genuine tables. However, their work starts with the identified tables without any detail about how to detect the table boundary in a document page. David et al. [19] compared the experimental results of extracting the tables from plain-text government statistical reports using Conditional Random Fields (CRF) and Hidden Markov Models (HMM) respectively. Unfortunately, many adopted line labels (e.g., separator) in [19] are too specific to be applicable in other document medium (e.g., HTML, PDF).. In general, [12] , recall is more important than precision because once the false negative table lines are removed, it is difficult to retrieve them back. However, the false positive rate can be easily lowered in later table structure decomposition step. In this paper, we propose a novel but effective method to quickly locate the boundary of a table by taking advantage of the aforementioned property. We also propose an exclusive based method for identifying table lines, which generates high recall and saves substantial effort to analyze the noisy lines. To our best knowledge, there are no proposed works that compare the machine learning based methods with the rule-based methods on the table boundary detection. In this paper, we apply two machine learning methods (CRF and SVM) on the table boundary detection Furthermore, we elaborate the feature selection, analyze the factor effects of different features, and compare the performance of CRF/SVM approaches with our proposed rulebased method.. Instead of extracting tables from the HTML documents [23] or the plain-text documents [19] , we focus on Portable Document Format (PDF). PDF is a widely used document format in digital libraries because it can preserve the appearance of the original document. Although a good number of researches have been done to discover the document layout by converting the PDFs to other types of files (e.g., image, html, text) in the past two decades, automatically identifying the document logical structures information (e.g., words, text lines, paragraphs, etc) and extracting the document components (e.g., figures, tables, mathematical formulas, etc) as well as the content [2] are still a challenging problem. The major reasons are as follows: 1) the structural information is not explicitly marked up because of the un-tagged nature of PDF format; 2) the text sequences are often messily generated by the existing PDF-to-text tools; 3) new noises can be generated by some necessary tools (e.g., OCR), if converting the PDFs into other media (e.g., image).. The rest of the paper is organized as follows. Section 2 reviews several relevant studies in table boundary detection area and the applied machine learning methods in this field. Section 3 introduces the sparse-line property of the table lines. Section 4 describes in detail the sparse line detection and the noise line removing using the conditional random field and support vector machine (SVM) techniques. We elaborate the label types and the feature sets. Section 5 explains the line construction before the line labeling. Section 6 explains how to locate the table boundary based on the labeled lines as well as the table keywords. The detailed experimental results are displayed in Section 7. We conclude our paper with plans for future work in Section 8. In this paper, we propose a novel method to detect the table boundary. Because most tables are text-based, we claim that the text object of PDF provides enough information for table detection. Within the text object, we believe that the font size is not so reliable as other work stated. Based on the sparse-line nature of tables, we propose a fast but effective method to detect the table boundary by only processing the sparse lines in a document page. Processing the sparse lines solely can also improve the performance of the text sequence resorting problem. Combining different keywords, this method is applicable to detect other document components, e.g., references or headers/footers. Our next tasks include applying the machine learning methods on the table structure decomposition field and the table classification field. Moreover, extending our test data sets and building a general table benchmark database are also our future works.

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


Memory modeling for system synthesis We present our methodology for developing models of cin-chip SRAM memory organizations. The models were created to enable the quick evaluation of energy, area, aind performance of different memory configurations considered during synthesis. The models are defined in terms of parameters, such as size and mode of operation, which are known at synthesis time. Our methodology does not require know1 edge of the underlying memory circuitry and provides imodels with average percentage errors within 8%. We found that only 10 different memories from a large span of possible memory sizes are needed to obtain reasonably accurate models, with average errors within 15%. We further use these models to evaluate different low power memory organizations and have seen energy reductions of up to 88 % . In this paper we present our modelirig methodology, discuss the important aspects in dtweloping the models, and show results of using the models in evaluating low power memory organizations.Power consumption of digital systems has become a critical design parameter. Ex tending battery life in portable applications and reducing cooling requirements in higher transistor density applications make power reduction a crucial consideration dui-ing digital system design.An important class of digital systems include applications, such as video image processing and speech recognition, which are extremely memory-intensive. In such systems, a significant amount of power is consumed during memory accesses. Thus, utilizin<: low-power memory organizations can greatly reduce the overall power consumption of the system. This work targets on-chip memories created by memory module generators in which there are many possible memory organizations in terms cif size, architecture, technology, etc. 'To utilize low power memory configurations during synthesis, we need models to quickly evaluate memory energy, area, and performance. These models need to be in lerms of parameters, such as size, organization, and mode of operation, which are known during synthesis time as opposed to lower level parameters such as extracted Permission to make digital or haid copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or io redistribute to lists, requires prior specific permission and/or a fee. Our models were developed to predict energy, delay, and area across the different possible sizes and organizations produced by memory module generators and for different modes of memory operations. Our modeling uses a simulation-based approach which enables the development of black box models. Unlike analytical models, simulationbased approaches do not require detailed knowledge of the underlying circuits, just basic input/output timing information which can be provided by the memory generator. The models are in terms of high-level parameters and can be easily used during synthesis. Our approach is similar to [9] but is generalized enough to handle more complex memory organizations and more modes of memory operation with higher accuracy.The focus of this paper is not just on our modeling methodology, but on defining the parameters and simulations necessary in building accurate models quickly and easily and showing how a synthesis tool can take advantage of these models in creating low power memory configurations. We have presented our modeling methodology for memory energy, area, and delay. Our methodology provides an easy and accurate way to develop memory models without detailed knowledge of the underlying circuitry. The models developed using our technique had average percentage errors within 8%. Using a weighted stepwise linear regression technique to determine the form of the models reduced the standard error over an order of magnitude from a simplified model approach. We showed that the size parameters were the most important to consider while developing the models and that it is only necessary to simulate 10 different sized memories to obtain models with average errors within 15%.. Memory architectural decisions are capable of profoundly moving the powerlareddelay characteristics of the design. Through such decisions we showed reductions of memory read energy of up to 88%. These complex decisions can be explored automatically within a synthesis system. Therefore, accurate and easy to develop memory models in terms of high-level parameters are necessary to explore a rich set of energy, area, and delay trade-offs.

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 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 B 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 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 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 B 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 B 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 O O O O O 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 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 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 B 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B O O O O O O O O O O O O O O O O O O O


The number of convex polyominoes and the generating function of Jacobi polynomials Lin and Chang gave a generating function of convex polyominoes with an m + 1 by n + 1 Some (binomial coefficients) identities arise from alternative solutions of combinatorial problems and incidentally give added significance to doing problems the "hard" way.-J. Riordan A polyomino is a connected union of squares in the plane whose vertices are lattice points. A polyomino is called convex if its intersection with any horizontal or vertical line is either empty or a line segment. Any convex polyomino has a minimal bounding rectangle whose perimeter is the same as that of the polyomino. Delest and Viennot [4] found a generating function for counting convex polyominoes by perimeter and showed that the number of convex polyominoes with perimeter 2n + 8, for n ≥ 0, is (2n + 11)4 n − 4(2n + 1) 2n n .. Later, Lin and Chang [9] gave a generating function for the number of convex polyominoes with an (m + 1) × (n + 1) minimal bounding rectangle, and Gessel [6] showed that their result implies that the number of such polyominoes is. which is easily seen to give a refinement of Delest and Viennot's formula.. Since Gessel [6] (see also Bousquet-Mélou [2] ) derived (2) from the generating function of Lin and Chang [9] (see also Bousquet-Mélou and Guttman [3] ), it would be interesting to find an independent proof of (2). For Delest and Viennot's formula (1) such a proof was already given by Kim [8] . The aim of this paper is to provide such a proof for (2) by generalizing Kim's elementary approach. It turns out that the resulting binomial coefficients identities are related to the generating function of Jacobi polynomials.. In the next section, we translate the enumeration of convex polyominoes with fixed minimal bounding rectangle as that of two pairs of non intersecting lattice paths, which results to evaluate a quadruple sum of binomial coefficients. In Section 3, we establish some binomial coefficients identities which lead to the evaluation of the desired sums.

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


Composite kernel learning The Support Vector Machine (SVM) is an acknowledged powerful tool for building classifiers, but it lacks flexibility, in the sense that the kernel is chosen prior to learning. Multiple Kernel Learning (MKL) enables to learn the kernel, from an ensemble of basis kernels, whose combination is optimized in the learning process. Here, we propose Composite Kernel Learning to address the situation where distinct components give rise to a group structure among kernels. Our formulation of the learning problem encompasses several setups, putting more or less emphasis on the group structure. We characterize the convexity of the learning problem, and provide a general wrapper algorithm for computing solutions. Finally, we illustrate the behavior of our method on multi-channel data where groups correpond to channels. Kernel methods have been extensively used in learning problems (Schölkopf & Smola, 2001) . In these models, the observations are implicitly mapped in a feature space via a mapping Φ : X → H, where H is a Reproducing Kernel Hilbert Space (RKHS) with reproducing kernel K : X × X → R.. We address the problem of learning the kernel in Support Vector Machines (SVM) and related methods. Indeed, the kernel is crucial in many respects, and its relevance is essential to the success of kernel methods. Formally, the primary role of K is to define the evaluation functional in H: ∀f ∈ H, f (x) = f, K(x, ·) H , but K also defines (i) H Our approach to solve Problem (10) draws on the MKL algorithm of Rakotomamonjy et al. (2007) . We use the wrapper scheme described below, where the outer loop is carried out by a projected gradient descent update.

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Probe threshold and probe trivially perfect graphs AbstractAn undirected graph G = (V, E) is a probe C graph if its vertex set can be partitioned 8 into two sets, N (non-probes) and P (probes) where N is independent and there

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Global convergence of an SQP method without boundedness assumptions on any of the iterative sequences Abstract Usual global convergence results for sequential quadratic programming (SQP) algorithms with linesearch rely on some a priori assumptions about the generated sequences, such as boundedness of the primal sequence and/or of the dual sequence and/or of the sequence of values of a penalty function used in the linesearch procedure. Different convergence statements use different combinations of assumptions, but they all assume boundedness of at least one of the sequences mentioned above. In the given context boundedness assumptions are particularly undesirable, because even for non-pathological and well-behaved problems the associated penalty functions (whose descent is used to produce primal iterates) may not be bounded below for any value of the penalty parameter. Consequently, boundedness assumptions on the iterates are not easily justifiable. By introducing a very simple and computationally cheap safeguard in the linesearch procedure, we prove boundedness of the primal sequence in the case when the feasible set is nonempty, convex, and bounded. If, in addition, the Slater condition holds, we obtain a complete global convergence result without any a priori assumptions on the iterative sequences. The safeguard consists of not accepting a further increase of constraints violation at iterates which are infeasible beyond a chosen threshold, which can always be ensured by the proposed modified SQP linesearch criterion. Consider the nonlinear programming problem minimize f (x) subject to x ∈ D = {x ∈ n | g i (x) ≤ 0, i = 1, . . . , m},. where f : n → and g : n → m are differentiable. One of the classical, yet practically efficient, approaches to solving (1) is the sequential quadratic programming (SQP) method (see, e.g., [1, Chap. 4] and [4, Part III] ). The development and analysis of the method go back to [12] and to [10, 11, 8, 5, 6, 9] , as well as numerous subsequent references.. Given some current primal iterate x k ∈ n , SQP computes the primal direction of change d k ∈ n as the solution of. where H k is an n × n symmetric positive definite matrix. (For the purposes of this discussion, we assume that D k = ∅. We note that this is always the case when g is convex and D = ∅. Also, having in mind global convergence properties, we shall not discuss any specifics concerning the choice of H k . Those specifics are much more relevant for local convergence rates.) In one of the most common classes of SQP methods, the next iterate x k+1 is computed by a linesearch procedure in the obtained direction d k , in order to decrease the value (with respect to x k ) of some nonsmooth penalty function. where β k is the current penalty parameter and the measure p of constraints violation can be given, for example, by. To ensure that d k is a direction of descent for ψ β k at x k , the standard way is to choose. where µ k ∈ m + is a Lagrange multiplier associated to the solution d k of (2) . Ideally (and to be really satisfactory from the mathematical standpoint), a global convergence result should show that the iterates approach stationary points of the problem under reasonable assumptions on the problem data (perhaps boundedness of some level sets or of the feasible set, perhaps some convexity, etc.). Despite rich literature and widely acknowledged practical usefulness, this is not the case for SQP. To prove global convergence of the method outlined above, some of the following a priori assumptions on the iterative sequences, in this or that combination, are made in the literature:. Assuming (A3), the assumption (A2) becomes automatic, of course (for β fixed). Furthermore, (A3) goes a long way to help justify (A1) in some situations, as will be discussed below. However, assuming (A2) or (A3) cannot be regarded really satisfactory in the context of SQP for the following reason. It is perfectly possible that for non-pathological and well-behaved problems,. no matter what β > 0 is. Therefore, as {x k } is constructed by descent steps for ψ β k , assuming (A2) or (A3) cannot be considered justifiable in this context. Consider, for example, the problem minimize x 3 subject to x 2 ≤ 1.. Evidently, ψ β (x) = x 3 + β max{0, x 2 − 1} is unbounded below for any β > 0. Hence, given that each SQP iteration is a descent step for ψ β k , one cannot take the possibility of lim k→∞ ψ β k (x k ) = −∞ lightly. This is quite disturbing, since the problem above satisfies just about any reasonable assumption one may want: we would like to compute a stationary point of a polynomial function on a feasible set which is compact, convex and satisfies the Slater condition. Hardly anything more could be asked. However, to the best of our knowledge, there is no result in the literature which could claim convergence of SQP for the presented example by looking at the problem data only, at least when it comes to the basic SQP scheme outlined above. We shall give a theoretical justification of why SQP works for problems like (6) .. Before proceeding, we survey some of the typical global convergence statements for basic (i.e., without complex modifications) SQP methods.. In [4, Th. 15.2] , the following situations are listed as possible for global behaviour of SQP (we simplify the statement by assuming that the problem data is defined on the whole space and that (1) has stationary points):. (i) {β k } is unbounded, in which case so is {µ k }; (ii) β k = β > 0 for all k ≥ k 0 , and one of the following holds. Clearly, out of all the possibilities, only the last one is satisfactory: it means that the residual of the Karush-Kuhn-Tucker (KKT) optimality conditions for problem (1) goes to zero, i.e., the iterates approach the set of stationary points of (1) . Therefore, to claim convergence, (A1) and (A2) need to be assumed or established, in order to eliminate the situations (i) and (ii-a).. There are many other SQP global convergence statements in the literature, but it seems that all of them (again, at least for simple schemes) use some of (A1)-(A3), perhaps proving the rest, in order to eliminate the unsatisfactory outcomes (i) and (ii-a). We shall cite only a few more. In [1, Prop. 4.14(a)], (A1) is assumed and the statement deals with accumulation points of {x k } without proving their existence (therefore, (A3) is needed for the result not to become vacuous). As already mentioned, (A1) is actually not difficult to guarantee under reasonable assumptions, if (A3) is assumed.. An example of this is [1, Prop. 4.14(b)], where g is convex and satisfies the Slater condition ∃x ∈ n such that g i (x) < 0, i = 1, . . . , m.. Under convexity of g, (7) and (A3), convergence follows. When, in addition to convexity of g and the Slater condition (7), the feasible set D is compact and f is bounded below on n , (A3) had been established in [6, Th. 3.3] (assuming also (A1) and using approximate minimization of ψ β in the direction d k instead of an implementable linesearch). Of course, boundedness below of f on n is just another way of assuming (A2). It should be noted here that while boundedness below of f on the feasible set D is a natural assumption necessary for existence of solutions of the problem, boundedness of f on n is a completely artificial requirement for a constrained problem. Summarizing, (A2) is still needed as an assumption even in the presence of (A1).. Let us go back to example (6) . The possible trouble, of course, could come from points. In particular, from the point. k points to the feasible set D of (1). It can be further seen that for. We note that the latter property is independent of the value of the stepsize α k which produced x k+1 , and in particular, independent of the value of β k chosen for the penalty function ψ β k . The above observation indicates that the primal iterates are getting closer to the bounded feasible set D, which makes the iterates bounded. And this is just due to the structure of SQP constraints and regardless of any other details of the method. From this, convergence of SQP when applied to the example (6) comes as no surprise.. What has been observed for the example (6) is not accidental, of course. As is easily seen, the directional derivative of p in the SQP direction d k ∈ D k is strictly negative at any infeasible point x k ∈ D. Indeed, for arbitrary x ∈ n and d ∈ n , we have that. where. This indicates that when we make a reasonably small step in the direction d k from an infeasible point, then the measure of infeasibility p is "likely" to decrease (because of (9)). Of course, the latter is not quite automatic if we ignore in the course of linesearch the values of p and look at ψ β k only. But we can guarantee the needed property by combining the linesearch for ψ β k with the linesearch for p. Actually, the only essential modification we shall make to the standard SQP method consists in ensuring that. where the infeasibility thresholdp ≥ 0 is a user-chosen parameter. Descent is not even required. The idea is that this safeguard of not accepting an increase of constraints violation at points which are already infeasible beyond a chosen (could be large) threshold, should be enough to make {x k } bounded under reasonable assumptions about the problem.. We should mention that the idea of limiting infeasibility of SQP iterates is certainly not new. Related conditions with similar goals can be found, e.g., in [7] , and possibly other publications as well, but in the context of significantly modified SQP schemes. Our goal here is to produce a complete global convergence result for the classical SQP method, perhaps very slightly modified.. Recall that assuming regularity of constraints (e.g., the Slater condition (7) in the case when g is convex), stationary points of (1) are characterized by the KKT conditions in variables (x, µ) ∈ n × m , which are

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