The local–global conjecture for scheduling with nonlinear cost
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Abstract
We consider the classical scheduling problem on a single machine, on which we need to schedule sequentially n given jobs. Every job j has a processing time \(p_j\) and a priority weight \(w_j\), and for a given schedule a completion time \(C_j\). In this paper, we consider the problem of minimizing the objective value \(\sum _j w_j C_j^\beta \) for some fixed constant \(\beta >0\). This nonlinearity is motivated for example by the learning effect of a machine improving its efficiency over time, or by the speed scaling model. For \(\beta =1\), the wellknown Smith’s rule that orders job in the nonincreasing order of \(w_j/p_j\) gives the optimum schedule. However, for \(\beta \ne 1\), the complexity status of this problem is open. Among other things, a key issue here is that the ordering between a pair of jobs is not well defined, and might depend on where the jobs lie in the schedule and also on the jobs between them. We investigate this question systematically and substantially generalize the previously known results in this direction. These results lead to interesting new dominance properties among schedules which lead to huge speed up in exact algorithms for the problem. An experimental study evaluates the impact of these properties on the exact algorithm A*.
Keywords
Scheduling Single machine Nonlinear cost function Pruning rules Algorithm A*1 Introduction
In a typical scheduling problem we have to order n given jobs, each with a different processing time, so to minimize some problem specific cost function. Every job j has a positive processing time \(p_j\) and a priority weight \(w_j\). A schedule is defined by a ranking \(\sigma \), and the completion time of job j is defined as \(C_j := \sum _i p_i\), where the sum ranges over all jobs i such that \(\sigma _i\le \sigma _j\). The goal is to produce a schedule that minimizes some cost function involving the jobs’ weights and the completion times.
A popular objective function is the weighted average completion time \(\sum w_jC_j\) (omitting the normalization factor 1 / n). It has been known since the 1950’s that optimal schedules are precisely the orders following an decreasing Smithratio \(w_j/p_j\), as has been shown by a simple exchange argument (Smith 1956).
In this paper, we consider the more general objective function \(\sum w_j C_j^\beta \) for some constant \(\beta >0\), and denote the problem by \(1\sum w_j C_j^\beta \). Several motivations have been given in the literature for this objective. For example it can model the standard objective \(\sum w_jC_j\), but on a machine changing its execution speed continuously. This could result from a learning effect, or the continuous upgrade of its resources, or from a wear and tear effect, resulting in a machine which works less effective over time. A recent motivation comes from the speed scaling scheduling model. In Dürr et al. (2014), and Megow and Verschae (2013) the problem of minimizing total weighted completion time plus total energy consumption was studied, and both papers reduced this problem to the problem considered in this paper for a constant \(1/2\le \beta \le 2/3\). However as we mention later in the paper, most previous research focused on the \(\beta =2\) case, as the objective function then represents a trade off between maximum and average weighted completion time.
2 Dominance properties
The complexity status of the problem \(1\sum w_j C_j^\beta \) is open for \(\beta \ne 1\) in the sense that neither polynomial time algorithms nor NPhardness proofs are known. For \(\beta =1\) the problem is polynomial, as has been shown by a simple exchange argument. When i, j are adjacent jobs in a schedule, then the order ij is preferred over ji whenever \(w_i/p_i > w_j/p_j\) and that is independent from all other jobs. In this case we denote this property by \(i\prec _\ell j\). Assume for simplicity that all jobs k have a distinct ratio \(w_k/p_k\), which is called the Smithratio. Under this condition \(\prec _\ell \) defines a total order on the jobs, that leads to the unique optimal schedule.
For general \(\beta \) values, the situation is not so simple, as in term of objective cost the effect of exchanging two adjacent jobs depends on their position in the schedule. So for two jobs i, j none of \(i\prec _\ell j, j\prec _\ell i\) might hold, which is precisely the difficulty of this scheduling problem.
However it would be much more useful if for some jobs i, j we know that an optimal schedule always schedules i before j, no matter if they are adjacent or not. This property is denoted by \(i\prec _g j\). Having many pairs of jobs with such a property could dramatically help in improving exhaustive search procedures to find an exact schedule. Section 8 contains an experimental study on the impact of this information on the performance of some search procedure.

Sen–DileepanRuparel (Sen et al. 1990) for any \(\beta >0\), if \(w_i>w_j\) and \(p_i\le p_j\), then \(i\prec _g j\).

Mondal–Sen–Höhn–Jacobs1 (Höhn and Jacobs 2012a) for \(\beta =2\), if \(w_i/p_i > \beta w_j/p_j\), then \(i\prec _g j\).

Mondal–Sen–Höhn–Jacobs2 (Höhn and Jacobs 2012a) for \(\beta =2\), if \(w_i\ge w_j\) and \(w_i/p_i > w_j/p_j\), then \(i\prec _g j\).
2.1 Related work
Embarrassingly, very little is known about the computational complexity of this problem, except for the special case \(\beta =1\) which was solved in the 1950’s (Smith 1956). In that case scheduling jobs in order of decreasing Smithratio \(w_j/p_j\) leads to the optimal schedule.
Two research directions were applied to this problem, approximation algorithms and branch and bound algorithms. The former has been proposed for the even more general problem \(1\sum f_j(C_j)\), where every job j is given an increasing penalty function \(f_j(t)\), that does not need to be of the form \(w_j t^\beta \). A constant factor approximation algorithm has been proposed by Bansal and Pruhs (2010) based on a geometric interpretation of the problem. The approximation factor has been improved from 16 to \(2 +\epsilon \) via a primaldual approach by Cheung and Shmoys (2011). The simpler problem \(1\sum w_j f(C_j)\) was considered by Epstein et al. (2010), who provided a \(4+\epsilon \) approximation algorithm for the setting where f is an arbitrary increasing differentiable penalty function chosen by the adversary after the schedule has been produced. A polynomial time approximation scheme has been provided by Megow and Verschae (2013) for the problem \(1\sum w_j f(C_j)\), where f is an arbitrary monotone penalty function.
Finally, Höhn and Jacobs (2012c) derived a method to compute the tight approximation factor of the Smithratio schedule for any particular monotone increasing convex or concave cost function. In particular, for \(f(t)=t^\beta \) they obtained for example the ratio 1.07 when \(\beta =0.5\) and the ratio 1.31 when \(\beta =2\).
Concerning branch and bound algorithms, several papers give sufficient conditions for the global order property, and analyze experimentally the impact on branch and bound algorithms of their contributions. Previous research focused mainly on the quadratic case \(\beta =2\), see Townsend (1978), Bagga and Karlra (1980), Sen et al. (1990), Alidaee (1993), Croce et al. (1993), and Szwarc (1998). Mondal and Sen (2000) conjectured that \(\beta =2 \wedge (w_i\ge w_j) \wedge (w_i/p_i > w_j/p_j)\) implies the global order property \(i\prec _g j\), and provided experimental evidence that this property would significantly improve the runtime of a branchandprune search. Recently, Höhn and Jacobs (2012a) succeeded to prove this conjecture. In addition they provided a weaker sufficient condition for \(i\prec _g j\) which holds for any integer \(\beta \ge 2\). An extensive experimental study analyzed the effect of these results on the performance of the branchandprune search.
3 Our contribution
All previously proposed sufficient conditions for ensuring that \(i\prec _g j\) were rather ad hoc, and are much stronger than what seems to be necessary. So this motivated our main goal of obtaining a precise characterization of \(i\prec _g j\), for each value \(\beta >0\).
In contrast, the condition \(i\prec _\ell j\) is fairly easy to characterize, using simple algebra, as has been described in the past by Höhn and Jacobs (2012a) for \(\beta =2\). This characterization holds in fact for any value of \(\beta \) and for completeness we describe it in Sect. 5.
As \(i\prec _g j\) trivially implies \(i\prec _\ell j\), the strongest (best) possible result one could hope for is that \(i \prec _g j\) occurs precisely when \(i\prec _\ell j\). If true, this would give to a local exchange property a broader impact on the structure of optimal schedules, and have a strong implication on the effect of nonlocal exchanges.
Having observed the optimal solutions of a large set of instances, this property seems to be the right candidate for a characterization. Moreover, this was also suggested by previous results for particular cases. For example, Höhn and Jacobs (2012a) showed that if \(\beta =2\) and \(p_i \le p_j\) then \(i\prec _g j\) if and only if \(i\prec _\ell j\). The same characterization has been shown for a related objective function, where one wants to maximize \(\sum w_j C_j^{1}\) (Vásquez 2014).
This situation motivates us to state the following conjecture.
Conjecture 1
(Local–Global Conjecture) For any \(\beta >0\) and all jobs i, j, \(i\prec _g j\) if and only if \(i\prec _\ell j\).
We succeed to show this claim in the case \(\beta \ge 1\). Somewhat surprisingly, the proof turns out to be extremely subtle and involved. In particular, it requires the use of several nontrivial properties of polynomials and carefully chosen inequalities among them, and then finally combining them using a carefully chosen weighted combination. Our proof distinguishes the cases \(p_j< p_i\) and \(p_j \ge p_i\). The first case is substantially easier than the second one. In fact, in the first case we can show that local–global conjecture for every \(\beta >0\). However, for the second case (\(p_j \ge p_i\)) when \(0<\beta <1\), we only give a necessary condition for \(i\prec _g j\).
4 Technical lemmas
This section contains several technical lemmas used in the proof of our main theorems.
Lemma 1
Proof
Some of our proofs are based on particular properties which are enumerated in the following lemma.
Lemma 2
 1.
\(f(x) \ge 0\) for \(x\ge 0\).
 2.
f is convex and nondecreasing, i.e., \(f^{\prime },f^{\prime \prime } \ge 0\).
 3.
\(f^{\prime }\) is logconcave (i.e., \(\log (f^{\prime })\) is concave), which implies that \(f^{\prime \prime }/f^{\prime }\) is nonincreasing. Intuitively this means that f does not increase much faster than \(e^x\).
 4.For every \(b>0\), the function \(g_b(x)= f(b+e^x)f(b)\) is logconvex in x. Intuitively this means that \(f(b+e^x)f(b)\) increases faster than \(e^{cx}\) for some \(c>0\). Formally this meansis increasing in y.$$\begin{aligned} y f^{\prime }(b+y)/\big (f(b+y)f(b)\big ) \end{aligned}$$(1)
The proof is based on standard functional analysis and is omitted.
Lemma 3

is decreasing in a and decreasing in b for any \(\beta >1\)

and is increasing in a and increasing in b for any \(0<\beta <1\).
Proof
The previous lemma permits to show the following corollary.
Corollary 1
Proof
Lemma 4
Proof
5 Characterization of the local order property
To simplify notation, throughout the paper we assume that no two jobs have the same processing time, weight, or Smithratio (weight over processing time). The proofs extend to the general case by considering an additional tiebreaking rule between jobs with identical parameters. For convenience we extend the notation of the penalty function f to the makespan of schedule S as \(f(S):=f\big ( \sum _{i\in S} p_i \big )\). Also we denote by F(S) the cost of schedule S.
Lemma 5

If \(p_i>p_j\) and \(\beta >1\), then \(\phi _{ij}\) is strictly increasing.

If \(p_i<p_j\) and \(\beta >1\), then \(\phi _{ij}\) is strictly decreasing.

If \(p_i>p_j\) and \(\beta <1\), then \(\phi _{ij}\) is strictly decreasing.

If \(p_i<p_j\) and \(\beta <1\), then \(\phi _{ij}\) is strictly increasing.
Proof
Lemma 6
For any jobs i, j, we have \(\lim _{t\rightarrow \infty } \phi _{ij}(t) = p_i/p_j.\)
Proof
These two lemmas permit to characterize the local order property, see Fig. 1.
Lemma 7

\(\beta > 1\) and \(p_i\le p_j\) and \(w_j/w_i\le \phi _{ji}(0)\) or

\(\beta > 1\) and \(p_i\ge p_j\) and \(w_j/w_i\le p_j/p_i\) or

\(0<\beta < 1\) and \(p_i\le p_j\) and \(w_j/w_i\le p_j/p_i\) or

\(0<\beta < 1\) and \(p_i\ge p_j\) and \(w_j/w_i\le \phi _{ji}(0)\).
6 The global order property
In this section, we characterize the global order property of two jobs i, j in the convex case \(\beta >1\), and provide sufficient conditions on the concave case \(0\le \beta <1\). Our contributions are summarized graphically in Fig. 1.
6.1 Global order property for \(p_i\le p_j\)
In this section, we give the proof of the conjecture in case i has processing time not larger than j. Intuitively this seems the easier case, as exchanging i with j in the schedule AjBi makes jobs from B complete earlier. However, the benefit of the exchange on these jobs cannot simply be ignored in the proof. A simple example with \(\beta = 2\) shows why this is so. Let i, j, k be 3 jobs with \(p_i=4,w_i=1,p_j=8,w_j=1.5,p_k=1,w_k=0\). Then \(i\prec _\ell j\), but exchanging i, j in the schedule jki increases the objective value, as \(F(ikj)=4^2 + 1.5\,\times \,13^2 = 269.5\) while \(F(jki)=1.5\,\times \, 8^2 + 13^2=265\). Now if we raise \(w_k\) to 0.3, then we obtain an interesting instance. It satisfies \(F(jki)<F(jik)\) and jki is the optimal schedule, but it cannot be shown with an exchange argument from ikj without taking into account the gain on job k during the exchange.
Theorem 1
The implication \(i\prec _\ell j \Rightarrow i\prec _g j\) holds when \(p_i\le p_j\).
Proof
The proof holds in fact for any increasing penalty function f. Let A, B be two arbitrary job sequences. We will show that the schedule AjBi has strictly higher cost than one of the schedules AijB, AiBj.
6.2 Global order property for \(\beta >1\)
Theorem 2
The implication \(i\prec _\ell j \Rightarrow i\prec _g j\) holds when \(\beta \ge 1\).
Proof
By Theorem 1 it suffices to consider the case \(p_j<p_i\). Assume \(i\prec _\ell j\) and consider a schedule S of the form AjBi for some job sequences A, B.
6.3 Global order property for \(0<\beta <1\) and \(p_j\le p_i\)
Theorem 3
The implication \(i\prec _\ell j \Rightarrow i\prec _g j\) holds when \(p_j\le p_i\), \(w_i/w_j \ge (p_i/p_j)^{2\beta }\), and \(0<\beta <1\).
Proof
7 Generalization
We can provide some technical generalizations of the aforementioned theorems. For any pair of jobs i, j, and job sequence T of total length t, we denote by \(i\prec _{\ell (t)}j\) the property \(F(Tij) < F(Tji)\). Now suppose that none of \(i\prec _\ell j\) or \(j\prec _\ell j\) holds, and say \(p_i > p_j\) and \(\beta >1\). Then from Lemma 5 it follows that there exists a unique time t, such that for all \(t^{\prime } < t\) we have \(i\prec _{\ell (t^{\prime })}j\) and for all \(t^{\prime } > t\) we have \(j\prec _{\ell (t^{\prime })}i\). These properties are denoted, respectively, by \(i\prec _{\ell [0,t)} j\) and \(j\prec _{\ell (t,\infty )} j\). In case \(p_i < p_j, \beta >1\) or \(p_i > p_j, 0<\beta <1\), we have the symmetric situation \(j\prec _{\ell [0,t)} i\) and \(i\prec _{\ell (t,\infty )} j\).
This notation can be extended also to the global order property. If for every job sequences A, B with A having total length at least t we have \(F(AiBj) < F(AjBi)\), then we say that i, j satisfy the global order property in the interval \((t,\infty )\) and denote it by \(i\prec _{g(t,\infty )}j\). The property \(i\prec _{g[0,t)}j\) is defined similarly for job sequences A, B of total length at most t.
The proof of Theorem 2 actually shows the stronger statement: if \(\beta >1\) and \(p_i\ge p_j\), then \(j\prec _{\ell (t,\infty )} i\) implies \(j\prec _{g(t,\infty )} i\). The same implication does not hold for interval [0, t), as shown by the following counter example. It consists of a 3job instance for \(\beta =2\) with \(p_i=13,w_i=7,p_j=8,w_j=5,p_k=1,w_k=1\). For \(t=19/18\), we have \(i\prec _{\ell [0,t)}j\) and \(j\prec _{\ell (t,\infty )}i\). But the unique optimal solution is the sequence jki, meaning that we do not have \(i\prec _{g[0,t)}j\) (Table 1).
Summary of generalized local–global property
\(p_i\le p_j\)  \(p_i\ge p_j\)  

\(0<\beta <1\)  \(\beta >1\)  
\(i \prec _{\ell [0,t)} j \Rightarrow i \prec _{g[0,t)} j\)  Yes  No 
\(j \prec _{\ell (t,\infty )} i \Rightarrow j \prec _{g(t,\infty )} i\)  Open  Yes 
8 Experimental study
We conclude this paper with an experimental study, evaluating the impact of the proposed rules on the performance of a search procedure. The experiments are based on a C++ program executed on a GNU/Linux machine with 3 Intel Xeon processors, each with 4 cores, running at 2.6 Ghz and 32 Gb RAM. In order to be independent on the machine environment, we measured the number of generated search nodes rather than running time. Hence we use a timeout which is not expressed in seconds, but in time units corresponding to the processing of a search node by the program. Note that we use the rules that we have proved (not the ones in the conjecture). Following the approach described in Höhn and Jacobs (2012a), we consider the Algorithm A* by Hart et al. (1972).
The search space is the directed acyclic graph consisting of all subsets \(S\subseteq \{1,\ldots ,n\}\). Note that the potential search space has size \(2^n\) which is already less than the space of the n! different schedules. In this graph for every vertex S there is an arc to \(S\backslash \{j\}\) for any \(j\in S\). It is labeled with j, and has cost \(w_j t^\beta \) for \(t=\sum _{i\in S} p_i\). Every directed path from the root \(\{1,\ldots ,n\}\) to the target \(\{\}\) corresponds to a schedule of an objective value being the total arc cost.
The algorithm A* finds a shortest path from the root to the target vertex, and as Dijkstra’s algorithm uses a priority queue to select the next vertex to explore. But the difference of A* is that it uses as weight for vertex u not only the distance from the source to u, but also a lower bound on the distance from u to the destination. A set S is maintained containing all vertices u for which a shortest path has already been discovered. Initially \(S=\{s\}\) for the root vertex s. In Dijkstra’s algorithm the priority queue contains all remaining vertices v, with the priority \(\min _{u\in S} d(s,u)+w(u,v)\), where w(u, v) is the weight of the arc (u, v). However in the algorithm A* this priority is replaced by \(\min _{u\in S} d(s,u)+w(u,v) + h(v)\), where h is some lower bound on the distance from v to the target. This function should satisfy \(h(v)=0\) if v is the target and \(h(u) \le w(u,v)+h(v)\) for every arc (u, v). The function h used in our experiments satisfies these properties.

Arc pruning The arc from S to \(S\setminus \{j\}\) for \(j\in S\) is pruned if \(i\prec _{\ell (t_1p_j)}j\), because placing job j adjacent to i at this position would be suboptimal.

Vertex pruning All arcs leaving vertex S are pruned, if there is a job \(j\in S\) with \(i \prec _{g[0,t_1]} j\), as again placing job j somewhere before job i would be suboptimal.
In a search tree such an arc pruning would cut the whole subtree attached to that arc, but in a directed acyclic graph (DAG) the improvement is not so significant. As the typical indegree of a vertex is linear in n, a linear number of arccut is necessary to remove a vertex from the DAG.
A simple additional pruning could be done when remaining jobs to be scheduled form a trivial subinstance. By this we mean that all pairs of jobs i, j from this subinstance are comparable with the order \(\prec _{\ell [0,t_1]}\). In that case the local order is actually a total order, which describes in a simple manner the optimal schedule for this subinstance. In that case we could simply generate a single path from the node S to the target vertex \(\{\}\). However, experiments showed that detecting this situation is too costly compared with the benefit we could gain from this pruning rule.
8.1 Random instances
We adopted their model for other values of \(\beta \) as follows. When \(\beta >1\), the condition for \(i\prec _g j\) of our conditions can be approximated, when \(p_j/p_i\) tends to infinity, by the relation \(w_i/p_i \ge \beta w_j/p_j\). Therefore, in order to obtain a similar “hardness” of the random instances for the same parameter \(\sigma \) for different values of \(\beta >1\), we choose the Smithratio according to \(2^{N(0,\beta ^2\sigma ^2)}\). This way the ratio between the Smithratios of two jobs is a random variable from the distribution \(2^{2N(0,\beta ^2\sigma ^2)}\), and the probability that this value is at least \(\beta \) depends only on \(\sigma \).
However when \(\beta \) is between 0 and 1, the our condition for \(i\prec _g j\) of our rule can be approximated when \(p_j/p_i\) tends to infinity by the relation \(w_i/p_i \ge 2 w_j/p_j\), and therefore we choose the Smithratio of the jobs according to the \(\beta \)independent distribution \(2^{N(0,4\sigma ^2 )}\).
The instances of our main test sets are generated as follows: For each choice of \(\sigma \in \{0.1,0.2,\ldots ,1\}\) and \(\beta \in \{0.5,0.8,1.1,\ldots ,3.2\}\), we generated 25 instances of 20 jobs each. The processing time of every job is uniformly generated in \(\{1,2,\ldots ,100\}\). Then the weight is generated according to the above described distribution. Note that the problem is independent from scaling of processing time or weights, motivating the arbitrary choice of the constant 100.
8.2 Hardness of instances
The results depicted in Fig. 5 confirm the choice of the model of random instances. Indeed the hardness of the instances seems to depend only little on \(\beta \), except for \(\beta =2\) where particularly strong precedence rules have been established. In addition the impact of our new rules is significant, and further experiments show how this improvement influences the number of generated nodes, and therefore the running time. Moreover it is quite visible from the measures that the instances are more difficult to solve when they are generated with a small \(\sigma \) value.
8.3 Comparison between forward and backward approaches
In this section, we consider a variant of the algorithm. The algorithm described so far is called the backward approach, and the variant is called the forward approach. Here a partial schedule describes a prefix of length t of a complete schedule and is extended to its right along an edge of the search tree, and in this variant the basic lower bound is \(h(S) := \sum _{i\in S} w_i (t+p_i)^\beta \). However in the backward approach, a partial schedule S describes a suffix of a complete schedule and is extended to its left. For this variant, we choose \(h(S) := \sum _{i\in S} w_i p_i^\beta \). Kaindl et al. (2001) give experimental evidence that the backward variant generates for some problems less nodes in the search tree, and this fact has also been observed by Höhn and Jacobs (2012a).
Later on, when we measured the impact of our rules in the subsequent experiments, we compared the behavior of the algorithm using the most favorable variant dependent on the value of \(\beta \) as described above.
8.4 Timeout
During the resolution a timeout was set, aborting executions that needed more than a million nodes. In Fig. 4 we show the fraction of instances that could be solved within the limited number of nodes. From these experiments we measure the instance sizes that can be efficiently solved, and observe that this limit is of course smaller when \(\sigma \) is small, as the instances become harder. But we also observe that with the usage of our rules much larger instances can be solved.
When \(\beta \) is close to 1, and instances consist of jobs of almost equal Smithratio, the different schedules diverge only slightly in cost, and intuitively one has to develop a schedule prefix close to the makespan, in order to find out that it cannot lead to the optimum. However for \(\beta =2\), the Mondal–Sen–Höhn–Jacobs conditions make the instances easier to solve than for other values of \(\beta \), even close to 2. Note that we had to consider different instance sizes, in order to obtain comparable results, as with our rules all 20 job instances could be solved.
8.5 Improvement factor
We observe that the improvement factor is more significant for hard instances, i.e., when \(\sigma \) is small. From the figures it seems that this behavior is not monotone, for \(\beta =1.1\) the factor is less important with \(\sigma =0.1\) than with \(\sigma =0.3\). However this is an artifact of our pessimistic measurements, since we average only over instances which could be solved within the time limit, so in the statistics we filtered out the really hard instances.
9 Performance measurements for \(\beta =2\)
For \(\beta =2\), Höhn and Jacobs (2012a) provide several test sets to measure the impact of their rules in different variants, see Höhn and Jacobs (2012b). For completeness we selected two data sets from their collection to compare our rules with theirs.
The first set called setn contains for every number of jobs \(n=1,2,\ldots ,35\), 10 instances generated with parameter \(\sigma =0.5\). This test set permits to measure the impact of our rules as a function of the instance size.
10 Performance depending on input size
11 Conclusion
We formulated the local–global conjecture for the single machine scheduling problem of minimizing \(w_j C_j^\beta \) for any positive constant \(\beta \). We proved it for \(\beta \ge 1\) substantially extending and improving over previous partial results. We also show some partial results for the remaining case \(0<\beta <1\).
We conducted experiments and measured the impact of our conditions on the running time (number of generated nodes) by an A*based exact resolution. Improvements by a factor up to 1e4 have been observed.
Based on extensive experiments we believe that the conjecture should also hold in this case. However, it seems to be substantially more complicated and new analytical techniques seem to be necessary. We also describe a more general class of functions for which our results hold. Determining the class of objective functions for which the local–global conjecture holds would also be a very interesting direction to explore.
Notes
Acknowledgments
We are grateful to the anonymous referees who spotted errors in previous versions of this paper. This paper was supported by the PHC Van Gogh grant 33669TC, the FONDECYT grant 11140566, the NWO grant 639.022.211 and the ERC consolidator grant 617951.
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