I Introduction
Motivation. Consider an operator of an electricity microgrid optimizing the joint production schedules of renewable and thermal power plants in order to satisfy, at each time period, the consumption constraints of its consumers. To optimize the costs and the renewables integration, this operator relies on demand response techniques, that is, taking advantage of the flexibilities of some of the consumers electric appliances—those which can be controlled without impacting the consumer’s confort, as electric vehicles or water heaters [1]. However, for privacy reasons, consumers are not willing to provide neither their consumption constraints nor their consumption profiles to a central operator or any third party, as this information could be used to induce private information such as their presence at home.
The global problem of the operator is to find an allocation of power (aggregate consumption) at each time period (resource) , such that (feasibility constraints of power allocation, induced by the power plants constraints). Besides, this aggregate allocation has to match an individual comsumption profile for each of the consumer (agent) considered. The problem can be written as follows:
(1a)  
(1b)  
(1c) 
The (aggregate) allocation can be made public, that is, revealed to all agents. However, the individual constraint set and individual profiles constitute private information of agent , and should not be reavealed to the operator or any third party.
The approach adopted in our paper is to deal with the problem (1) as two kinds of interdependent subproblems. The firsts are optimal resource allocation problems, or master problems, which consists in finding an aggregate allocation over resources (. The second kind is problems of finding the disaggregation of a given aggregate allocation , that is, to find an individual profile for each agent (consumer) satisfying her individual constraint (1b), such that the aggregate of the profiles is the optimal allocation (1c) determined in a master problem.
Aside from the example above, ressource allocation problems (optimizing common resources shared by multiple agents) find many applications in energy [2, 1], logistics [3], distributed computing [4], health care [5] and telecommunications [6]. In these applications, several entities or agents (e.g. consumers, stores, tasks) share a common resource (energy, products, CPU time, broadband) which has a global cost for the system. For large systems composed of multiple agents, the dimension of the overall problem can be prohibitive and one can rely on decomposition and distributed approaches [7, 8, 9] to answer to this issue. Besides, agents’ individual constraints are often subject to privacy issues [10]. These considerations have paved the way to the development of privacypreserving, or nonintrusive methods and algorithms, e.g. [11, 12].
In this work, we consider that each agent has a global demand constraint (e.g. energy demand or product quantity), which confers to the disaggregation problem the particular structure of a transportation polytope [13]: the sum over the agents is fixed by the aggregate solution , while the sum over the resources are fixed by the agent global demand constraint. Besides, individual constraints can also include minimal and maximal levels on each resource, as for instance electricity consumers require, through their appliances, a minimal and maximal power at each time period.
Main Results. The major contribution of the paper is to provide a nonintrusive and distributed algorithm (Algorithm 4) that computes an aggregated resource allocation , optimal solution of the—possibly nonconvex—optimization problem (1), along with feasible individual profiles for agents, without revealing the individual constraints of each agent to a third party, either another agent or a central operator. The algorithm solves iteratively intances of master problem by constructing successive approximations of the aggregate feasible set of (1) for which a disaggregation exists, by adding a new constraint on to , before solving the next master problem.
To identify whether or not disaggregation is feasible and to add a new constraint in the latter case, our algorithm relies on the alternating projections method (APM) [14, 15] for finding a point in the intersection of convex sets. Here, we consider the two following sets: on the one hand, the affine space defined by the aggregation to a given resource profile, and on the other hand, the set defined by all agents individual constraints (demands and bounds). As the latter is defined as a Cartesian product of each agent’s feasibility set, APM can operate in a distributed fashion. The sequence constructed by the APM converges to a single point if the intersection of the convex sets is nonempty, and it converges to a periodic orbit of length otherwise. Our key result is the following: if the APM converges to a periodic orbit, meaning that the disaggregation is not feasible, we construct from this orbit a polyhedral cut, i.e. a linear inequality satisfied by all feasible solutions of the global problem (1), but violated from the current resource allocation (Thm. 4). Adding this cut to the master problem, we can recompute a new resource allocation and repeat this procedure until disaggregation is possible. Another major result stated in this paper is the explicit upper bound on the convergence speed of APM in our framework (Thm. 2), which is obtained by spectral graph theory methods, exploiting also geometric properties of transportation polytopes. This explicit speed shows a linear impact of the number of agents, which is a strong argument for the applicability of the method in large distributed systems.
Related Work. A standard approach to solve resource allocation problems in a distributed way is to use a Lagrangian (dual) decomposition technique [8, 16, 17]. ^{inline,disable}^{inline,disable}todo: inline,disable
SG: I rewrote lagrange *decomposition* and not *relaxation*. An important point is whether any earlier paper has implemented Lagrange decomposition in a privacypreserving way, using the same kind of SMC technique. If so, we should be more specific. If this has not been done explicitly, it would be better to say something like: “Lagrange decomposition is generally used to reduce the dimension. It may also be implemented in a way which preserve privacy, in a distributed way (see Remark XXX in Section IV – add a remark there with a comparison between Lagrange relaxation and our method.). However, Lagrange decomposition methods are based on strong duality property, requiring global convexity hypotheses which are not satisfied in many practical problems: for instance, in the field of energy, the master allocation problem is generally a mixed integer linear program (MILP), see
Sec. V. Those techniques are generally used to decompose a large problems into several subproblems of small dimension. They may also be implemented in a way which preserve privacy (see Remark 2 in Sec. IV). However, Lagrangian decomposition methods are based on strong duality property, requiring global convexity hypothesis which are not satisfied in many practical problems (e.g. MILP, see Sec. V). On the contrary, our method can be used when the master allocation problem is not convex. In [2], the authors study a disaggregation problem similar to the one considered in this paper. Their results concern zonotopic sets, which is different from the structure we described in Sec. II. The APM has been the subject of several works in itself [15, 18, 19]. The authors of [20]provide general results on the convergence rate of APM for semialgebraic sets. They show that the convergence is geometric for polyhedra. However, it is generally hard to compute explicitly the geometric convergence rate of APM, as this requires to bound the singular values of certain matrices arising from the polyhedral constraints. In
[21], the authors provide an explicit convergence rate for APM on a class of polyhedra arising in submodular optimization. The sets they consider differ from the present transportation polytopes.Structure. In Sec. II, we describe the master resource allocation problem and formulate the associated disaggregation problem. In Sec. III, we focus on the APM and state our main results. In Sec. IV, we apply these results to describe a nonintrusive version of APM (NIAPM) that is used to describe our nonintrusive algorithm for computing an optimal resource allocation. Finally, in Sec. V, we provide a concrete numerical example based on a MILP to model the management of a local electricity system (microgrid), and study numerically the influence of the number of agents on the time needed for convergence of our algorithm.
Notation.Vectors and matrices are denoted by bold fonts, denotes the transpose of , denotes the vector of size ,
stands for the uniform distribution on
. We use to denote the Frobenius norm , and to denote the Euclidean projection on a convex set .Ii Master problem and disaggregation structure
^{inline,disable}^{inline,disable}todo: inline,disableSG: I rewrote the next para, this has still to be improved, these sentences are decisive in the readingIn this work, we suppose an operator wishes to determine an allocation of resources, represented by a dimensional vector , in order to minimize a global cost function , for instance, an electricity power economic dispatch (or the allocation of different types of merchandise in warehouses in logistics applications) subject to a set of constraints described by a feasibility set . This problem can be nonconvex either because of nonconvex costs or because of a nonconvex feasible set (see Sec. V). In the proposed method, the operator will consider master problems of the form:
(2a)  
s.t.  (2b) 
where the set is an aggregate approximation of disaggregation constraints. Indeed, the resource allocation has to be shared between agents (e.g. consumers). Each agent has a global demand (total energy needed) and some lower and upper bounds on each of the resource . The admissible set of profiles of agent is therefore:
(3) 
The disaggregation problem consists in finding individual profiles of a given aggregated allocation such that is feasible for each agent :
(4)  
where 
Following (4), the disaggregated profile refers to , while the aggregated profile refers to the allocation .
Problem (4) may not always be feasible. Some necessary conditions for a disaggreagation to exist, obtained by summing the individual constraints on , are the following aggregated constraints:
(5) 
However, (inline,disable) are not sufficient conditions, as shown in Fig. 1 where the problem (4) is represented as a flow or circulation problem from source nodes to sink nodes .
Indeed, with this circulation representation of the disaggregation problem (4), an immediate consequence of Hoffmann theorem [22, Thm. 3.18][23] is the following characterization of the disaggregation feasibility, which involves an exponential number of inequalities:
Theorem 1.
The disaggregation problem (4) is feasible (i.e. ) iff for any :
(6) 
The inequality (6) has a simple interpretation: the residual demand (the left hand side composed of demand and exports minus production) in cannot exceed the import capacity (right hand side of the inequality). One can see that, in the example of Fig. 1, inequality (6) does not hold when using the cut composed of the dashed nodes and .
There are two main reasons for which solving (1) is harder than solving (2) and (4) separately:

[wide,label=)]

the dimension of (1) can be huge, as the number of agents can be really important, for instance in the example of individual consumers;

also, and this is the main motivation of this work, the information related to and might not be available to the centralized operator in charge of optimizing resources , as this information may be confidential and kept by each agent , not willing to reveal it to any third party.
In the next sections, we provide a method that addresses those two issues, by considering subproblems (2) and (4) independently and iteratively, and exploiting the decomposable structure of problem (4).
Iii Alternate Projection Method (APM)
Iiia Convergence of APM on Transportation Polytopes
In this section, we consider a fixed aggregated profile and present the Von Neumann Alternate Projections Method (APM) [14] which solves the problem Eq. 4 of finding a point in the intersection . In the remaining, we will often ommit and just write to denote . The key idea of the method proposed in this paper is to use results of APM to generate a cut in the form of (6) and to add it as a new constraint in the master problem (2) to “improve” the aggregated profile for the next iteration. As described in Algorithm 1, APM can be used to decompose (4) and only involves local operations.
The convergence of Algorithm 1 is proved by Thm. 2:
Theorem 2 ([15]).
Let and be two convex sets with bounded, and let and be the two infinite sequences generated by Algorithm 1 with . Then there exists and such that:
(7a)  
(7b) 
In particular, if , then and converge to a same point .
If disaggregation is not feasible, Thm. 2 states that APM will “converge” to an orbit of period 2.
The convergence rate of APM has been the subject of several works [18, 20], and it strongly depends on the structure of the sets on which the projections are done: for instance, if the sets are polyhedral, [20, Prop. 4.2] shows that the convergence is geometric. However, there are very few cases in which an explicit upper bound on the convergence rate has been proved. In our case, we are able to obtain such a bound, as shown in the following theorem:
Theorem 3.
Proof.
Appendix B provides a sketch of the proof.
Thm. 3 shows that the APM is efficient in our case of bounded transport polytopes. It shows that the number of iterations for a given accuracy grows linearly in the number of agents .
As stated in (4), the set is a Cartesian product , so that the projection (13) can be computed by projections on , which can be executed in parallel. Now, instead of solving the quadratic program by standard interior point methods and due to its particular structure, we can use the algorithm of Brucker [24], which has a complexity in . On the other hand, is a projection on an affine space, and the solution can be obtained explicitly as:
(8) 
IiiB Generation of a cut from APM iterates
Our key result is the following: in the case where APM converges to a periodic orbit with (see Thm. 2), we obtain from an inequality (6) that is violated by :
Theorem 4.
Proof.
Appendix A gives the sketch of the proof of Thm. 4. The complete proofs will be given elsewhere.
One can see that, intuitively, is the subset associated to that minimizes the right hand side of (6). Note that Thm. 4 gives an alternative constructive proof of Hoffman circulation’s theorem (Thm. 1) in the case of a bipartite graph of the form of Fig. 1. Moreover, in the case where the disaggregation problem (4) is not feasible, the negation of equation (11) provides a new valid constraint as a condition for the existence of a disaggreagated profile of . This constraint can be used in the master problem (2) to update the vector of resources for the next iteration. This constraint only involves the aggregate information on the users profile. To make the process fully nonintrusive, we explain in Sec. IVA how the operator can compute this constraint without making the agents reveal their profiles .
Iv Nonintrusive Projections and Computation of Disaggregated Optimal Resources
Iva NonIntrusive Alternate Projections Method (NIAPM)
Because of the particular structure of the problem, the projections in APM can be computed separately by the operator and the agents. The projection is made by the operator, which only requires to know and the aggregate profile according to from (8). The projection on is executed in parallel by each agent: computes which only needs her private information and . However, in the way APM is described in Algorithm 1, the operator and the agents still need to exchange the iterates at each step. To avoid the transmission of agents’ profiles to the operator, we use a secure multiparty computation (SMC) technique (see [25]) which enables the operator to obtain the aggregate profile in a nonintrusive manner, as described in Algorithm 2.
The main idea of SMC is that, instead of sending her profile , agent splits for each into random parts , according to an uniform distribution and summing to (Lines 34). Thus, each part taken individually does not reveal any information on nor on , and can be sent to agent . Once all exchanges of parts are completed (Line 5), and has herself received the parts from other agents, agent computes a new aggregate quantity (Line 8), which does not contain either any information about any of the agents, and sends it to the operator (Line 9). The operator can finally compute the quantity .
Remark 1.
As and are random by construction, an eavesdropper aiming to learn the profile of has no choice but to intercept all the communications of to all other agents (to learn and ) and to the operator (to learn ). To increase the confidentiality of the procedure, one could use any encryption scheme (such as RSA [26]) for all communications involved in Algorithm 2.
We can use this nonintrusive computation of aggregate in APM to obtain a nonintrusive algorithm NIAPM (Algorithm 3) in which agents do not reveal neither their profiles nor their constraints to the operator.
One can see that and computed in Lines 4 and 9 in Algorithm 3 correspond to the projections computed in the original APM Algorithm 1. In Algorithm 3, the operator obtains the aggregate profile (Line 6), computes and sends the corrections to all agents (Line 7). Then, each agent can compute locally the projection by applying the correction (Line 9).
Using (8), we get . Thm. 4 uses this limit value through . Yet, from APM, one can only access to and thus to the approximation , computed on Line 17), where is a predefined constant. However, we show that for small enough and a wellchosen value of , we obtain , so that we get the termination result:
Proposition 1.
For , Algorithm 3 terminates in finite time.
The termination of the loop Lines 212 is ensured by Thm. 3. In the case where , Algorithm 3 terminates. Otherwise, if , then Algorithm 3 terminates (i.e. Line 19 is True and a new cut is found) as soon as , where and with . The complete proof is ommited here.
In practice, we can start with a large to obtain the first constraints while avoiding useless computation, and then half if needed (Line 22) until the termination condition holds.
Remark 2.
SMC techniques could also be used to implement nonintrusive Lagrangian decomposition methods. However, these methods rely on a convexity hypothesis that we do not need in the proposed method.
IvB Nonintrusive Disaggregation of Optimal Allocation
In this section, we describe a method to compute a solution of the global problem (1), that is, an optimal resource allocation for which a disaggregation exists, along with an associated disaggregated profile for each agent . This computation is done in a nonintrusive manner: the operator in charge of does not have access neither to the bounding constraints and of the agents nor to the agents disaggregated profile , as detailed in Algorithm 4 below.
Algorithm 4 iteratively calls NIAPM (Algorithm 3) and in case disaggregation is not possible (Line 7), a new constraint is added (Line 9), obtained from the quantity defined in (11), to the resource problem (2). This constraint is an inequality on and thus does not reveal significant individual information to the operator. The algorithm stops when disaggregation is possible (Line 5). The termination of Algorithm 4 is ensured by the following property and the form of the constraints added (10):
Proposition 2.
Algorithm 4 stops after a finite number of iterations, as at most constraints (Line 9) can be added to the master problem (Line 3).
Although there exist some instances with an exponential number of independent constraints, this does not jeopardize the proposed method: in practice, the algorithm stops after a very small number of constraints added (see the example of Sec. V). Intuitively, we will only add constraints “supporting” the optimal allocation .
Thus, Algorithm 4 is a method which enables the operator to compute a resource allocation and the agents to adopt profiles , such that solves the global problem (1), and the method ensures that both:

the information relative to each agent constraints (upper bounds , lower bounds , demand );

the final disaggregated profile (as well as the iterates and in NIAPM)
are kept confidential by agent and can not be induced by a third party (either the operator or any other agent ).
V Application to Management of a Microgrid
We apply the proposed method to solve a nonconvex distributed problem in the energy field. We consider a microgrid [27] composed of N electricity consumers with flexible appliances (such as electric vehicles or water heaters), a photovoltaic (PV) power plant and a conventional generator.
Va Mixed Integer Problem Formulation
The operator responsible of the microgrid aims at satisfying the demand constraints of consumers over a set of time periods , while minimizing the energy cost for the community. We have the following characteristics:

[wide]

the PV plant generates a nondispatchable power profile at marginal cost zero;

the conventional generator has a starting cost , minimal and maximal power production , and piecewiselinear and continuous generation cost function :
where and ;

each agent has some flexible appliances which require a global energy demand on , and has consumption constraints on the total household consumption, on each time period , that are formulated with . These parameters are confidential because they could for instance contain some information on agent habits.
The master problem (2) can be written as a MILP (12):
(12a)  
(12b)  
(12c)  
(12d)  
(12e)  
(12f)  
(12g)  
(12h)  
(12i)  
(12j)  
(12k)  
(12l) 
In this formulation (12b12f) are a mixed integer formulation of the generation cost function : one can show that the boolean variable is equal to one iff for each . Note that only appears in (12a) because of the continuity assumption on .
Constraints (12g12h) ensure the on/off and starting constraints of the power plant, (12j) ensures that the power allocated to consumption is not above the total production, and (12k12l) are the aggregated feasibility conditions already referred to in (inline,disable). Note that more complex and realistic MILP models exist for power plants (e.g. [28]), but with the same structure than (12). The nonconvexity of (12) comes from the existence of starting costs and on minimal the power constraint, which makes necessary to use boolean state variables .
VB Parameters
We simulate the problem described above for different values of ,, ,, and one hundred instances with random parameters for each value of . A scaling factor is applied on parameters to ensure that production capacity is large enough to meet consumers demand. The parameters are chosen as follows:

[wide,labelindent=0pt]

(hours of a day);

production costs: , , , and ;

photovoltaic: for , otherwise (see Fig. 3);

for consumption parameters, we used , and , so that individual feasibility () is ensured.
# master pb.  193.6  194.1  225.5  210.9  194.0 

# projs.  9506.9  15366.7  24319.3  26537.5  26646.4 
VC A limited impact of the number of agents
We implement Algorithm 4 using Python 3.5. The MILP (12) is solved using Cplex Studio 12.6 and Pyomo interface. Simulations are run on a single core of a cluster at 3GHz. For the convergence criteria (see Lines 12 and 13 of Algorithm 3), we use with the operator norm defined by (to avoid the factor in the convergence criteria appearing with ), and starts with . The largest instances took around 10 minutes to be solved in this configuration and without parallel implementation. As the CPU time needed depend on the cluster load, it is not a reliable indicator of the influence on on the complexity of the problems. Moreover, one advantage of the proposed method is that the projections in APM can be computed locally by each agent in parallel, which could not be implemented here for practical reasons.
Tab. I gives two robust indicators of the influence of on the problem complexity: the number of master problems solved and the total number of projections computed, on average over the hundred instance for each value of :

[wide]

one observes that the number of master problems solved (MILP (12) ), which corresponds to the number of constraints or “cuts” added to the master problem, remains almost constant when increases;

in all instances, this number is way below the upper bound of possible constraints (see proof of Prop. 2), which suggests that only a polynomial number of constraints are added in practice;

the average total number of projections computed for each instance (total number of iterations of the while loop of Algorithm 3, Line 2 over all calls of APM in the instance) increases in a sublinear way, as illustrated in Fig. 2, which is even better that one could expect from the upper bound given in Thm. 3.
Vi Conclusion
We provided a nonintrusive algorithm that enables to compute an optimal resource allocation, solution of a–possibly nonconvex–optimization problem, and affect to each agent an individual profile satisfying a global demand and lower and upper bounds constraints. Our method uses local projections and works in a distributed fashion. Hence, it ensures that the problem is not affected by the high dimension relative to the large number of agents, and that it is privacypreserving, as agents do not need to reveal any information on their constraints or their individual profile to a third party. Several extensions and generalizations can be considered. First, we could generalize the abstract circulation problem on the bipartite graph depicted in Fig. 1 to an arbitrary network, where the set of nodes is partitioned in parts defining sets on which we could make alternating projections. Second, our method relies on the particular structure obtained from the form of constraints. Although these kind of constraints are widely used to model many practical situations, it would also be useful to obtain similar results for arbitrary (or polyhedral) agents constraints. Last, a deeper complexity analysis, with a thinner upper bound on the maximal number of constraints (cuts) added in the algorithm (see Prop. 2 and Tab. I) would constitute interesting results.^{disable}^{disable}todo: disableSG: the last problem is rather difficult theoretically, perhaps may be rephrased in a less ambitious way (study the complexity)
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Appendix A Proof of Prop. 3
To show Thm. 4, we formulate the projections and as the solutions of the constrained quadratic programs:
(13a)  
(13b)  
(13c) 
and:
(14a)  
(14b) 
where are the Lagrangian multipliers associated to the constraints. Although there is no such explicit characterization of the solution of (13) as the one (8) given for (14), we can obtain the following properties:
Proposition 3.
Suppose that and consider the sets and given by (9). Then we have the following:

[label=()]

and ;

;

;

the sets , and are nonempty.
PJ: Agreed, I’ll add that
Appendix B Proof of Thm. 2
For this analysis, we use the space , where the to coordinates correspond to agent , for . We make use of the following results:
Lemma 1 ([21]).
For APM on polyhedra and , the sequences and converge at a geometric rate, where the rate is bounded by the maximal value of the square of the cosine of the Friedrichs angle between a face of and a face of , where is given by:
Lemma 2 ([21]).
Let and be matrices with orthonormal rows and with equal numbers of columns and the set of singular values of . Then if , then . Otherwise, .
In our case, the polyhedra is an affine subspace with , where denotes the Kronecker product. The matrix has orthonormal rows and the direction of is .
Describing the faces is more complex. We have a polyhedral description of , and the faces of are subsets of the collection of affine subspaces indexed by (with ):
The associated linear subspace is given by , where the first rows of are given by , and the matrix has more rows, where , corresponding to the saturated inequalities or
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