Abstract
We consider mixedinteger optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixedinteger optimal control problem without combinatorial constraints and a mixedinteger problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sumuprounding strategies and mixedinteger linear programming techniques. The coupling is handled using a penaltyapproach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands.
1 Introduction
Optimal control problems subject to integer restrictions on some part of the controls have recently received a lot of attention in the literature. This problem class is a convenient way to model, for instance, autonomous driving in case of vehicles with gear shift power units [25], contact problems such as robotic multiarm transport [5], or the operation of networked infrastructure systems such as gas pipelines [19], water canals [20], traffic roads [14], and power transmission lines [13] with switching of valves, gates, traffic lights and interconnectors, respectively. Often, the integer controls are additionally constrained in order to prevent certain switching configurations, to limit the number of switches or to enforce certain dwell or dead times after a switch. In this paper, we consider such combinatorial constraints with a focus on conditions which cannot be imposed pointwise and hence couple over time.
A discretization of such problems naturally leads to mixedinteger nonlinear programs that often become computationally intractable when the discretization stepsizes tend to zero. Moreover, when passing to the limit, one may face convergence issues [22]. A computationally much more efficient alternative solution approach is based on decomposition techniques, splitting the problem into a continuous subproblem by partial outer convexification with relaxation of binary multipliers (POC) combined with a combinatorial integral approximation problem (CIAP) [21, 23, 24, 28, 34, 36, 37]. However, in the presence of combinatorial constraints that couple over time, this approach only yields feasible solutions with a priori lower bounds, but yet without any characterization of optimality in some reasonable sense on the mixedinteger level.
We consider here another approach based on the idea of alternating direction methods (ADM). This will provide feasible solutions which can be characterized as partially optimal in a lifted sense. The approach uses POC and CIAP in one direction and a mixedinteger linear problem (differing from the combinatorial integral approximation problem) in another direction. Both directions are weakly coupled using a penalty term with adapting an idea outlined in [11]. Based on exactness of this penalty, we provide a convergence result of this method. Our analysis applies to problems in the setting of abstract semilinear evolutions subject to control constraints. However, the methods can be extended to state constraints. In particular, the techniques apply to optimal control problems with ordinary differential equations. The method can be also seen as an adaptation of classical feasibility pump algorithms (for an overview, see [3]) with heavy structure exploitation for mixedinteger optimal control problems.
One feature of our approach is a clear separation of the combinatorial aspects from the continuous control aspects of the problem. This is in contrast to, e.g., the approach proposed in [38, 39]. There, the full problem is discretized and then a variant of an Alternating Direction Method of Multipliers (ADMM) is used to obtain heuristic solutions. Further recent applications of ADM type methods are related to electricity networks, see e.g. [4, 9, 27].
In a numerical study, we consider two problems from the mintOC.de library [35], which we augment by minimum dwelltime constraints. We compare the proposed approach with direct discretization and mixedinteger programming techniques in order to address local vs. global optimality and to the decomposition with POC and CIAP as a heuristic.
We note that continuous reformulations of such problems with switching time and mode insertion optimization or combinatorial constraints can also be seen as an alternating direction method [8, 29, 32, 33], but that the approach proposed here is different.
The article is organized as follows. In Sect. 2, we present the problem formulation. In Sect. 3, we extend the framework of alternating direction methods to partial \(\varepsilon \)optimality. In Sect. 4, we apply the \(\varepsilon \)ADM framework together with penalty techniques to mixedinteger optimal control problems as the main algorithm and develop the convergence theory. In Sect. 5, we provide our numerical results. In Sect. 6, we give concluding remarks.
2 Problem formulation
We consider a mixedinteger optimal control problem of the form
where for some \(M\in \mathbb {N}\) and \(p \in (0,\infty ]\), Y and U are Banach spaces, \(Y_t=C([0,T];Y)\), \(U_t=L^p(0,T;U)\), \(V_t=L^\infty (0,T;\{0,1\}^M)\), A is a (densely defined) linear operator on Y, f is a nonlinear mapping \(f: Y \times U \times \{0,1\}^M \rightarrow Y\), \(\Phi \) is a nonlinear function \(\Phi : Y \rightarrow \mathbb {R}\cup \{\infty \}\) representing state costs, \(\Sigma \) is a subset of \(U_t\) representing constraints on the continuous control u, and \(\Gamma \) is a subset of \(V_t\) representing combinatorial constraints (e. g., dwelltime constraints and switching order constraints).
We say that the set of combinatorial constraints \(\Gamma \) has a uniform finiteness property, if there exists a constant \(n_s \in \mathbb {N}\) such that \(v \in \Gamma \) implies that v is piecewise constant with at most \(n_s\) switching points.
Example 1
(Combinatorial constraints) With \(v=(v_1,\ldots ,v_M)\) being the componentwise respresentation of \(v \in V_t\) and the total variation of the ith component on the interval \((t_1,t_2) \subset (0,T)\) being denoted by
where \(C^1_c(t_1,t_2)\) denotes continuously differentiable vector functions of compact support on \((t_1,t_2)\), we can for example enforce a minimal dwelltime \(\tau _{\min }\) with the constraint
or directly limit the total number of switches for the ith component to \(n_s^{\max }\) by
In both cases it is easy to see that any set \(\Gamma \subset V_t\) containing one of these constraints has the uniform finiteness property. Further additional constraints are of course possible, for instance, a maximum dwelltime \(\tau _{\max }\) for a subset of components \(I \subset \{1,\ldots ,M\}\)
Constraints of the form (2)–(4) or variants of it are significant in many applications that involve switching control, but they are typically extremely difficult to be treated in the context of mixedinteger optimal control because they are not defined pointwise in time.
Concerning the wellposedness of the problem, we make the following assumptions.
Assumption 1
Suppose that A generates a strongly continuous semigroup \(e^{tA}\) on Y, and that there exists a constant \(K>0\) such that, for all \(v \in \{0,1\}^M\),

(i)
the map \((y,u) \mapsto f(y,u,v)\) is continuous on \(Y \times U\),

(ii)
\(\Vert f(y,u,v)\Vert _Y \le K(1+\Vert y\Vert )\) for all \(y \in Y\), \(u \in U\),

(iii)
\(\Vert f(y,u,v)f(z,u,v)\Vert _Y \le K\Vert yz\Vert _Y\) for all \(y,z \in Y\), \(u \in U\).
Moreover, assume that \(\Phi :Y \rightarrow {\mathbb {R}}\) is Lipschitz continuous on bounded subsets of Y.
We note that the conditions of Assumption 1 are sufficient for the state equation (1b) to admit a unique solution \(y(\cdot ;u,v)\) in C([0, T]; Y) given by
Of course, other conditions are also possible, see e.g., [30]. Moreover, the uniform finiteness property is crucial for the existence of optimal solutions for the problem (1). For additional problem specific assumptions, appropriate wellposedness results can for example be obtained via parametric programming. The next theorem illustrates this for the case of generators of immediately compact semigroups.
Theorem 1
Suppose that Assumption 1 holds. Moreover, assume that X is separable and reflexive, \(\Sigma =U_t\) and that \(f(y,U,v_i)\) is closed and convex in Y, \(e^{tA}\) is compact for \(t>0\) and that \(\Gamma \) has the uniform finiteness property. Then the problem (1) has an optimal solution \((y^*,u^*,v^*)\).
Proof
Under the uniform finiteness property, the problem (1) can be considered as a parametric two stage problem, where the inner problem consists of minimizing with respect to u and the outer problem is a minimization with respect to finitely many switching times \(\tau _k\). Under the given assumptions, the inner problem has an optimal solution and the optimal value depends continuously on the initial data [6], and hence via (5) on the switching times \(\tau _k \in [0,T]\). The claim then follows from the extreme value theorem of Weierstrass.\(\square \)
For the solution approach considered below, we note that under the Assumption 1, we can consider the reduced problem
and results for (6) can be carried over to the original problem (1) via (5).
3 ADM with \(\varepsilon \)optimality
As a solution approach we extend here the idea of ADM. Suppose we were to minimize a nonlinear function \(\varPsi (u,v)\) over \(u \in U\) and \(v \in V\) subject to constraints \((u,v) \in \Omega \) for some given feasible set \(\Omega \). Further suppose that we can compute \(\varepsilon \)optimal solutions for each of the partials u (with v fixed) and v (with u fixed). Then, given some \(\varepsilon \ge 0\) and some guess \((u^0,v^0)\), we can consider the following sequential approach to compute a solution candidate \((u^*,v^*)\):

(i)
Find \(u^{l+1}\) such that \(\varPsi (u^{l+1},v^l) \le \varPsi (u,v^l) + \frac{\varepsilon }{2}\) for all \((u,v^l) \in \Omega \).

(ii)
If \(\varPsi (u^{l+1},v^l) \ge \varPsi (u^l,v^l)\frac{\varepsilon }{2}\), set \((u^*,v^*)=(u^l,v^l)\).

(iii)
Find \(v^{l+1}\) such that \(\varPsi (u^{l+1},v^{l+1}) \le \varPsi (u^{l+1},v) + \frac{\varepsilon }{2}\) for all \((u^{l+1},v) \in \Omega \).

(iv)
If \(\varPsi (u^{l+1},v^{l+1}) \ge \varPsi (u^{l+1},v^l)\frac{\varepsilon }{2}\), set \((u^*,v^*)=(u^{l+1},v^l)\).

(v)
Set \(l=l+1\) and continue with step (i).
This algorithm may not terminate. For classical ADM, there are wellknown conditions under which we can ensure that the algorithm does not cycle, i.e. that the algorithm does not get stuck in a loop of different solutions; for a discussion, see [11]. However, if it terminates, we can conclude that \((u^*,v^*)\) satisfies
This can be seen as follows: If the algorithms terminates in step (ii), we have for some \({\hat{l}}\) from (ii)
and from (i)
hence, with \(v^{{\hat{l}}}=v^*\), we get
Moreover, from step (iii) with \(l={\hat{l}}1\), we have
hence, again with \(u^{{\hat{l}}}=u^*\), we have
If the algorithm terminates in step iv), we have for some \({\hat{l}}\) from iv)
and from (iii)
Hence, with \(u^{{\hat{l}}+1}=u^*\), we get
Moreover, from (i), we have
Hence with \(v^{{\hat{l}}}=v^*\), we get
We shall call points \((u^*,v^*)\) satisfying (7a) and (7b) p\(\varepsilon \)optimal as a shorthand for partially \(\varepsilon \)optimal.
4 ADM and pminima for mixedinteger optimal control problems
Concerning the mixedinteger optimal control problem (1) or equivalently for the reduced form (6), a natural ADM splitting is using the directions \(u \in U\) and \(v \in V\). However, in the direction of v, this still results in a mixedinteger nonlinear optimization problem subject to a differential equation. To avoid this, we will instead use that (1) is equivalent to
and consider a splitting with respect to the directions (u, v) and \(\tilde{v}\). This particular splitting is chosen deliberately in view of the fact that the two subproblems can be efficiently solved to \(\varepsilon \)optimallity with existing techniques. Motivated by (7a) and (7b), we say that a point \((u^*,v^*)\) is p\(\varepsilon \)minimal for (6) if \(([u^*,v^*],v^*)\) is p\(\varepsilon \)optimal for (8). Consistently, we say that a point \((y^*,u^*,v^*)\) is p\(\varepsilon \)minimal for the original problem (1) if \((u^*,v^*)\) is a p\(\varepsilon \)minimum of (6) and \(y^*\) is a solution of the state equation (1b) with \(u=u^*\) and \(v=v^*\). We note that p\(\varepsilon \)minima are not necessarily global \(\varepsilon \)minima. But any global minimum of (1) is p\(\varepsilon \)minimal with \(\varepsilon =0\). For brevity, we call p\(\varepsilon \)minima with \(\varepsilon =0\) just pminima.
The above discussion motivates to compute pminima of good quality. To this end, we enforce the coupling of v and \({\tilde{v}}\) in (8c) weakly with a suitable penalty term. The penalty parameter can then eventually be used to avoid getting stuck in p\(\varepsilon \)minima with too high objective. This idea was introduced recently in [11] for classical ADM in the context of feasibility pumps for MINLPs. Suitably adapted to our setting here, we are going to show an exactness result for the penalty problem.
We may consider the optimal value function of the reduced problem (6) partially with respect to u
as a function \(\eta :V_t\rightarrow \mathbb {R}\cup \{\infty ,+\infty \}\). We will impose the following technical assumption on \(\eta \) using the 1norm \(\mathchoice{\leftv \right}{v}{v\Vert }{v}_{l_1}=\sum _{i=1}^M v_i\) on \(\{0,1\}^M\).
Assumption 2
Given an optimal solution \((y^*,u^*,v^*)\) of (1), or equivalently an optimal solution \((u^*,v^*)\) of problem (6) the value function \(\eta \) defined in (9) is locally Lipschitz continuous in the sense that for all \(\delta >0\) there exists a constant L such that
for all \(v \in V_t\) with \(\int _0^T \mathchoice{\leftv^*(t)v(t) \right}{v^*(t)v(t)}{v^*(t)v(t)\Vert }{v^*(t)v(t)}_{l_1}\,dt \le \delta \).
Assumption 2 is typically satisfied if the optimal solution \((y^*,u^*,v^*)\) satisfies a constraint qualification. For instance for mixedinteger linear quadratic optimal control problems the Lipschitz continuity of the optimal value function under a constraint qualification of a Slatertype is discussed in [16]. For mixedinteger finitedimensional problems, conditions are provided in [15] and [22].
Now we consider the following auxiliary problem
with a penalty parameter \(\rho \ge 0\). With (5) and (6a) we can reduce (11) to
The following result shows the exactness of the penalty term in (11) and relates global minima of (6) to pminima of (12).
Theorem 2
Let \((u^*,v^*)\) be a global minimum of problem (6) satisfying Assumption 2. Then, there exists a penalty parameter \(\bar{\rho }\) such that \(([u^*,v^*], v^*)\) is a pminimum of (12) for all \(\rho \ge {\bar{\rho }}\).
Proof
Let \((u^{*}, v^{*})\) be an optimal solution of (6). We note that by construction \(u = u^{*}\), \(v = v^{*}\), \(\tilde{v} = v^{*}\) is a global minimum of (8).
We have to show that \(([u^*,v^*],{\tilde{v}}^*)\) satisfies
with \(\varepsilon =0\).
It can be seen that condition (13a) holds with \(\varepsilon =0\) for all \(\rho \ge 0\). This follows directly from the definition of \(\varPsi _{\rho }\).
To show condition (13b) with \(\varepsilon =0\), we assume we are given (u, v) and consider the triple \((u, v, {v}^{*})\). Without loss of generality, we may assume that u is chosen optimally in the sense of (9). By the definition of \(\varPsi _{\rho }\), condition (13b) with \(\varepsilon =0\) is equivalent to
We directly observe that if \(\varPsi (u,v) \ge \varPsi (u^*,v^*)\) holds, then claim (14) is true for all \(\rho \ge 0\). This is, for instance, the case if \(v = {v}^{*}\) holds, i.e. if \((u,v,{v}^{*})\) is feasible for (8), because \((u^{*}, v^{*})\) is a global minimum of (8). Hence, we only need to consider the case in which \(\varPsi (u,v) \le \varPsi (u^*,v^*)\) and \(v \ne {v}^{*}\) hold.
As we have chosen u optimally and because \(u^{*}\) is also an optimal choice for \(v^{*}\) in the sense of (9), we can rewrite (14) further to obtain
Now, we may use Assumption 2 to obtain
This shows that condition (15) is fulfilled for all \(\rho \ge L\). Hence, we have shown that condition (13b) holds with \(\varepsilon =0\) if we set \({\bar{\rho }} = L\). \(\square \)
The essential idea of the proposed method now is to solve (12) using the method discussed at the beginning of Sect. 3. So, in each iteration of the outer loop (index k) the penalty parameter \(\rho \) is increased. In the inner loop (index l), we apply an alternating direction method to (12) with this parameter \(\rho \) until we find a partial \(\varepsilon \)optimum. For this, we need to be able to solve two subproblems to accuracy \(\varepsilon \): (12) with fixed \(\tilde{v}\) and (12) with fixed (u, v). For fixed \(\tilde{v}\) (12) reduces to an optimal control problem and for fixed (u, v) (12) reduces to a mixedinteger linear problem (assuming the constraints describing \(\Gamma \) are linear). Both of these problem types can be solved to \(\varepsilon \) accuracy with standard techniques.
The algorithm is summarized in Algorithm 1.
Concerning the convergence of Algorithm 1, we can now make the following statements.
Theorem 3
Let \(\rho ^k \nearrow \infty \) and let \((u^k,v^k,{\tilde{v}}^k)_k\) be a sequence generated by Algorithm 1 with \((u^k,v^k,{\tilde{v}}^k) \rightarrow (u^*,v^*,{\tilde{v}}^*)\). Then \((u^*,v^*,{\tilde{v}}^*)\) is a pminimum of the feasibility measure \(\chi (v,{\tilde{v}})=\int \limits _0^T v(t){\tilde{v}}(t)_{l_1}\,dt\).
Proof
Let \((u,v,{\tilde{v}}^k)\) be feasible for (12). Then
Let \(\bar{\rho }\) be a cluster point of the sequence \(\left( \frac{\rho ^k}{\rho ^k}\right) _k\) and \((\rho ^l)_l\) be a subsequence for which \(\left( \frac{\rho ^l}{\rho ^l}\right) _l\) converges to \(\bar{\rho }\). Then, dividing the inequality (16) by \(\rho ^l\) yields
Taking the limit \(l \rightarrow \infty \) yields
An analogous inequality holds for any feasible \((u^k,v^k,{\tilde{v}})\). \(\square \)
Corollary 1
Let \(\rho ^k \nearrow \infty \) and let \((u^k,v^k,{\tilde{v}}^k)_k\) be a sequence generated by Algorithm 1 with \((u^k,v^k,{\tilde{v}}^k) \rightarrow (u^*,v^*,{\tilde{v}}^*)\) and let \((u^*,v^*,{\tilde{v}}^*)\) be feasible for (8). Then \((u^*,v^*)\) is pminimal for (6).
Proof
This follows from Theorem 3 and using that feasibility of \((u^*,v^*,{\tilde{v}}^*)\) for (8) implies \(v^*={\tilde{v}}^*\). \(\square \)
Note that in the inner loop of Algorithm 1, we compute p\(\varepsilon \)minima, but that Corollary 1 says that a feasible limit of a converging sequence generated by ADMSUR is a p\(\varepsilon \)minimum with \(\varepsilon =0\). Moreover, note that the two subproblems for (12) in Algorithm 1 can be solved efficiently. Finally, we note that \(\rho ^k \nearrow \infty \) is needed in Theorem 3 and Corollary 1, because Theorem 2 guarantees exactness of the penalty only in a global minimum. It must be observed that even in the finitedimensional case, there are only slightly stronger results known (see [11], Theorems 8 and 11). It is instructive to note that in the finitedimensional case the assumption of convexity immediately yields convergence to global optima and the assumption of differentiabilty convergence to local optima. To us, this indicates that the mixedinteger part of the problem makes it difficult to prove anything about convergence to local (or even global) optima for these types of methods.
For any fixed \(\tilde{v}\) the problem (12) is equivalent to the problem
with \(r_i\), \(i=1,\ldots ,\tilde{M}=2^M\), enumerating the configurations \(\{0,1\}^M\), \(Z_t=L^1(0,T)\) and \(V_t=L^\infty (0,T;\{0,1\}^{{\tilde{M}}})\). Letting \(({\bar{y}},\bar{z},{\bar{u}}, {\bar{w}})\) be a solution of (17) with the relaxation \(w \in L^\infty (0,T;[0,1]^{{\tilde{M}}})\) and \(w^n\in V_t\) be a sequence generated by the sumup rounding algorithm of [34, 36] and \(y_n\), \(z_n\) be the corresponding solutions of (17b) and (17c) with \(w=w^n\), then under Assumption 1
see [28] for details. Under additional assumptions on A and f, even error estimates are available [18, 21, 36]. In particular, (18) shows that this solution approach yields an \(\frac{\epsilon }{2}\)optimal solution for a sufficiently fine control grid. We refer to this solution approach for subproblem (17) as the (POC)step.
Further, for any fixed (u, v) the problem (12) reduces to
Here, standard quadrature rules and mixedinteger linear programming techniques can be used to compute an \(\frac{\varepsilon }{2}\)optimal solution again for a sufficiently fine control grid. We refer to this solution approach for the subproblem (19) as the (MIP) step.
The sumup rounding algorithm in the (POC)step can be interpreted as the solution of the following combinatorial integral approximation problem (CIAP)
for a piecewise constant function \(w^n\) on a fixed grid, see [37]. We therefore refer to the penalty\(\varepsilon \) method in Algorithm 1 as ADMSUR. An interesting variant of this Algorithm is to skip SUR in the POCstep, i.e., doing the step in the relaxed direction of w and using the MIPstep to recover integer feasibility. We refer to this variant as ADM (without SUR). For comparison, we also consider the heuristic to apply POC to the original problem formulation without combinatorial constraints and to recover a feasible solution via the following mixedinteger problem
again on a fixed grid for \(w^n\), see [21, 37]. We refer to this approach as CIAP. Note that in contrast to (19), the cost function in (21) is not a norm on \(V_t\). A convergence result for this subproblem in an ADM framework such as for the ADMSUR algorithm is therefore an open problem.
5 Numerical study
We test the proposed methods on two benchmark examples from the mintOC.de library [35], which we augment by minimum dwelltime constraints in order to prohibit infinitely many switching events. To model the dwell time condition we used the basic MIP constraints. These do not, however, form a complete description of the socalled min up/down polyhedron. One can either use a complete description in the original variable space [26], where additional constraints are separated with cutting planes or use an extended formulation [31] instead. As the main focus of this article is the solution quality, we stick to the basic formulation above.
Our computations are based on CasADi [1] for the model equations and their derivatives and the solvers Ipopt [40] for nonlinear programming problems and Gurobi [17] for quadratic and linear mixedinteger programs.
The ADM method was used with \(\rho = 10^{3}, 10^{2}, \dotsc , 10^6\) and \(\varepsilon = 10^{3}\). We terminated the method when the value of the penalty term dropped below a tolerance of \(10^{4}\), because increasing \(\rho \) beyond that point will not change the iterates much on a fixed discretization. At the end of this section, we study the dependency of the penalty parameter adaptation for smaller choices of the multiplicative increment. Our study confirms the experience from the finitedimensional case (the version described in [11]) that the effect of the penalty adaptation strategy does not change the qualitiative behavior of the method. In our point of view, several different strategies can be used to identify pminima with low objectives for example within a global search strategy like branchandbound. An interesting direction for further research seems to be the use of weighted 1normpenalties with adaption strategies for the weights as used in the finitedimensional case (see, for instance, [10, 12]). This is not straightforward in the infinitedimensional case, because that would make the strategy discretizationdependent.
5.1 Fuller’s problem
For our numerical study, we consider a variant of Fuller’s problem augmented with minimum dwelltime constraints
The problem is notoriously difficult, because the solution of the problem without dwelltime constraints (i.e., for \(\tau _{\min }=0\)) exhibits chattering [41].
We compare our proposed ADMbased method (with and without SUR) with a direct global MixedInteger Quadratic Programming (MIQP) method and CIAP. We discretize Equations (22b) and (22c) using a Gauss–Legendre collocation of degree 4 on an equidistant partition of [0, 1] with 200 collocation intervals. The same collocation nodes are also used for approximating the integral term in (22a) via Gauss–Legendre quadrature. The control discretization is piecewise constant, with jumps allowed only at the boundary of the collocation intervals but not at the collocation nodes. Because the objective (22a) is quadratic in y and the constraints (22b) and (22c) are linear in y, the same holds for their discretized counterparts. Hence, we obtain a discretized MIQP, which we solve, where possible, to global optimality using Gurobi. The resulting objective values and corresponding single CPU runtimes are depicted in Tables 1 and 2. It appears that the ADMbased methods have some advantage both in quality and runtime over CIAP for the instances with larger \(\tau _\mathrm {min}\), which are harder for POCbased heuristics (but appear to be simpler for the MIQP approach). Exemplary for two selected values dwelltimes \(\tau _{\min }\), the resulting state \(y_1\) in problem (22) is shown in Fig. 1.
5.2 Network of transmission lines
This problem was described in [13]. The telegraph equations are based on a \(2\times 2\) hyperbolic system of partial differential equations and describe the voltage and current on electrical transmission lines in time \(t\in [0,T]\) and space \(x\in [0,l]\). The state variable \(\varvec{\xi }(x,t)=(\xi ^+(x,t),\xi ^(x,t))\) represents right or lefttraveling components on each line of the network and is governed by
where \(\varvec{\Lambda }\) is a diagonal matrix including the speed of propagation in each direction and \(\mathbf {B}\) denotes a symmetric matrix with nonnegative entries. The dynamics on the lines are coupled at nodes via the boundary condition
The distribution matrices \(\mathbf {D}^{\pm }({\varvec{v}})\) depend on binaryvalued controls \({\varvec{v}}(t)\in \{0,1\}\), which are used to switch off specified connections in the network while the continuousvalued controls \({\varvec{u}}(t)\) denote the power generation at the producer nodes in the network, cf. Fig. 2. The goal is to minimize the quadratic deviation of the accumulated power delivery \(C_s(t,\varvec{\xi })=\sum _{r \in \delta _S} \xi ^+_r(l_r,t)\) (with \(\delta _S\) being the set of all lines adjacent to node s) from the demand \(Q_s(t)\) at the consumer nodes \(V_S\), i.e.,
This problem can be written in abstract form as
with A and \(B(v_i)\), \(i=1,\ldots ,M\) being unbounded linear operators on Hilbert spaces using abstract semigroup theory [2]. Though (26) is not of the form (1), the solution is still given by the variation of constants formula (5) with \(f(y,u,v)=B(v)u\), see e.g., [7].
For the computational experiments, we use the publicly available^{Footnote 1} Python implementation, which uses a classical upwinding Finite Volume discretization with 4 equidistant volumes per line with forward Euler timestepping with 104 equidistant time steps as in [13]. The minimum dwelltime constraints are set to \(\tau _\mathrm {min} = 1\).
The Figs. 3, 4 and 5 illustrate the results for a scenario, in which a small subgrid of the network can be islanded, see Fig. 2. We observe that the binary decisions can be partly equalized by reactions in the power generation at the producer nodes.
Finally, we present a numerical study of the influence of the penalty parameter adaptation on the resulting objective function in Table 4. To this end, we use a coarser discretization of the subgrid scenario (2 equidistant finite volumes per transmission line, 52 equidistant time steps) and increase the penalty parameter in multiplicative steps of \(\root i \of {10}\) for varying \(i \in \{ 1, 2, 4, 8 \},\) starting from \(\rho = 10^{3}\). The outer loop is terminated when the value of the penalty term drops below \(10^{4}\). We observe that the ADM without CIAP is largely unaffected by the penalty adaptation choice, while the ADM with CIAP shows a more pronounced dependence.
6 Conclusion
We conclude that the proposed penaltyADM method performs notably well for our benchmark problems within the class of mixedinteger optimal control problems with dwelltime constraints. The quality of the computed solutions outperforms the other considered heuristic solutions for large dwelltimes. We think that it is worthwhile to use this heuristic inside of exact methods to ensure that good feasible solutions are found early on in the solution process. Moreover, the convergence theory shows that the proposed method computes partial minima in a lifted sense. The comparison with a global solution for a full discretization shows that these partial minima are in general not global minima. However, we note that this is not surprising because we used a local solver for the POCstep. The proposed methods can be extended in various directions such as considerations of state constraints, mixedinteger corrector steps from linearizations and of course more general problem classes.
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Acknowledgements
The second and fourth author were supported by the Deutsche Forschungsgemeinschaft (DFG) within the Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks, Projects A03 and B07. The research of the fourth author has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government. The third author was supported by the German Federal Ministry for Education (BMBF) and Research under grants MOPhaPro (05M16VHA) and MOReNet (05M18VHA) while the first author was supported by the BMBF under grant ENets (05M18VMA).
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Göttlich, S., Hante, F.M., Potschka, A. et al. Penalty alternating direction methods for mixedinteger optimal control with combinatorial constraints. Math. Program. 188, 599–619 (2021). https://doi.org/10.1007/s10107021016569
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DOI: https://doi.org/10.1007/s10107021016569
Keywords
 Mixedinteger optimization
 Partial differential equations
 Dwelltime constraints
 Alternating direction methods
 Penalty methods
Mathematics Subject Classification
 49J15
 49J20
 65K05
 90C11