Combining discrete and continuous information for multi-criteria optimization problems

Teichert, Katrin; Seidel, Tobias; Süss, Philipp Article — Published Version Combining discrete and continuous information for multicriteria optimization problems Mathematical Methods of Operations Research Provided in Cooperation with: Springer Nature Suggested Citation: Teichert, Katrin; Seidel, Tobias; Süss, Philipp (2024) : Combining discrete and continuous information for multi-criteria optimization problems, Mathematical Methods of Operations Research, ISSN 1432-5217, Springer, Berlin, Heidelberg, Vol. 100, Iss. 1, pp. 153-173, https://doi.org/10.1007/s00186-024-00849-0 This Version is available at: https://hdl.handle.net/10419/314961 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Mathematical Methods of Operations Research (2024) 100:153–173 https://doi.org/10.1007/s00186-024-00849-0 ORIGINAL ARTICLE Combining discrete and continuous information for multi-criteria optimization problems Katrin Teichert1·Tobias Seidel1·Philipp Süss1 Received: 11 April 2023 / Revised: 4 January 2024 / Accepted: 4 January 2024 / Published online: 23 February 2024 © The Author(s) 2024 Abstract In multi-criteria optimization problems that originate from real-world decision making tasks, we often find the following structure: There is an underlying continuous, possibly even convex model for the multiple outcome measures depending on the design variables, but these outcomes are additionally assigned to discrete categories according to their desirability for the decision maker. Multi-criteria deliberations may then take place at the level of these discrete labels, while the calculation of a specific design remains a continuous problem. In this work, we analyze this type of problem and provide theoretical results about its solution set. We prove that the discrete decision problem can be tackled by solving scalarizations of the underlying continuous model. Based on our analysis we propose multiple algorithmic approaches that are specifically suited to handle these problems. We compare the algorithms based on a set of test problems. Furthermore, we apply our methods to a real-world radiotherapy planning example. Keywords Multi-criteria optimization ·Decision making ·Non-linear optimization · Pareto front approximation 1 Introduction In many practical applications, the decision maker’s utility function is of a step-wise nature even though the underlying measure and its dependence on the real-valued optimization variables is continuous. For example, in radiotherapy one tries to have a sufficiently high dose in the target and then, provided this is the case, as little dose in surrounding organs at risk as possible. While the dose values in those nearby organs are continuously dependent on administered radiation, from a physician’s point of BKatrin Teichert [email protected].de 1Fraunhofer Institute for Industrial Mathematics (ITWM), Fraunhoferplatz 1, 67663 Kaiserslautern, Germany 123 154 K. Teichert et al. view they are judged on whether they exceed certain threshold values associated with particular side effects. Thereby, a dose in an organ might be labeled as either “dangerously high”, “acceptable”, or “ideal”. The multi-criteria decision making would then deliberate on the trade-offs between different organs based on these broad categorizations. We formalize this observation by defining the following type of multi-criteria optimization problem: P:min x∈Md1(f1(x)),...,dk(fk(x)) (1) where we assume M⊆Rnto be compact and non-empty, f1,..., fk:M→R to be continuous functions, and d1,...,dk:R→Rto be monotone increasing. We let f:= (f1,..., fk)and d:= (d1◦π1,...,dk◦πk), where πidenotes the projection from a vector in Rkto the i-th component. We assume that there is no a priori preference between the objectives. To establish the discrete nature of our problem, we require the following assumption for the monotone increasing utility functions di: Assumption 1.1 (Discrete utility) For i∈{1,...,k}there is a finite set i⊂Rsuch that di:R→i. In the literature, there is a vast amount of general-purpose approaches that can be applied to multi-criteria problems with discrete and continuous decision variables and objectives alike (Steuer and Choo 1983; Benson and Sayin 1997; Das and Dennis 1998; Schandl et al. 2002). While these are applicable to P, they do not take into account its special structure. An efficient algorithm for Pshould ideally combine certain characteristics of algorithms intended specifically for continuous problems with those specific for discrete problems. When facing a problem with the structure of P, the aim is to find the non-dominated solutions with respect to the discrete utility function evaluations di. However, we also have an underlying continuous problem that we can potentially capitalize on: Pc:min x∈Mf1(x),..., fk(x). (2) Similar to how non-dominated solutions are obtained in the purely continuous setting, efficient solutions to (2) can be calculated with potentially very efficient general-purpose continuous solvers, such as IPOPT (Wächter and Biegler 2006)orknitro (Byrd et al. 2006). Suitable scalarization methods are weighted sum if the problem is convex, and -constraint (see e.g. Ehrgott 2005), or Pascoletti–Serafini scalarization (Pascoletti and Serafini 1984) in the case of non-convex problems. Particularly suited approaches for approximating the Pareto front of a continuous problem are sandwiching (Serna 2012; Bokrantz and Forsgren 2012) in case the problem is convex, and hypervolume or hyperboxing algorithms (Bringmann and Friedrich 2010; Teichert 2014) if the problem is non-convex. However, algorithms for continuous problems are approximation algorithms: they aim at calculating a discrete set of non-dominated solutions that represent the Pareto 123 Combining discrete and continuous information for… 155 front up to some approximation error measure, often by establishing lower and/or upper bounds on the front (Sayın 2000; Klamroth et al. 2002; Eichfelder 2009). In contrast, for our setting we do not aim at an approximation of the Pareto front, but a full representation. That is, we want to find at least one efficient solution for each non-dominated point in the image space of the discrete utility functions. In this regard, our approach is in line with many algorithmic approaches for calculating the Pareto front of a purely discrete problem (Ulungu and Teghem 1994; Holzmann and Smith 2018). While it is in fact possible to solve the multi-criteria problem Panalogously to a purely discrete problem with those techniques, such an approach is unnecessarily computationally expensive, as the more difficult discrete problem is solved repeatedly and not the potentially simpler continuous problem. On the other hand, one could solve the continuous problem Pcin the approximative sense and, in a second step, infer the discrete front. But then many points on the continuous Pareto front may be calculated that are redundant for the discrete Pareto front of P. This is why in this paper we aim to combine both worlds. We will present algorithms that avoid solving the discrete problem and rather solve the simpler continuous problem Pc. Simultaneously, the algorithms will incorporate the discrete aspects of Pto avoid solving redundant continuous problems. The outline for this paper is as follows. We first investigate the relationship between the solutions of Pand Pc(Sect.2). Based on this theoretical insight, we then propose different algorithms to find all non-dominated points for P(Sect.3). Finally, we demonstrate and evaluate these algorithms using a set of test problems and an example from radiotherapy (Sect.4). We conclude with a discussion of our results (Sect.5). 2 Theoretical results In this section, we investigate the theoretical properties of P(1). In particular, we describe the relationship between peculiar solutions of Pand their corresponding counterparts with respect to the underlying continuous problem Pc(2). This is crucial for the development and the analysis of the algorithms described later in this paper. Note that for all observations within this section, Assumption 1.1 is not required; they hold for arbitrary monotone functions. Before we present the theoretical results, we summarize some notation used throughout this paper. For two vectors y1,y2∈Rkwe write y1y2if the inequality y1 i≤y2 iholds for every component i=1,...k(analogously for ,<,>). We say that a point y∈Rkdominates a point y∈Rkif: yi≤y ifor all i∈{1,...,k} yi<y ifor at least one i∈{1,...,k}. We say that a point y∈Rkstrictly dominates a point y∈Rkif for all i∈{1,...,k} the inequality yi<y iholds. The image of the combined MCO problem Pis denoted by 123 156 K. Teichert et al. Y:= (d◦f)(M). (3) For the the underlying continuous problem Pcthe image is denoted by Yc:= f(M). (4) A point x∈Mis called efficient solution for P(for Pc)ifthereisnoy∈Y(y∈Yc) such that ydominates (d◦f)(x)(dominates f(x)). The corresponding image point is called non-dominated. A point x∈Mis called weakly efficient solution for P(for Pc)ifthereisnoy∈Y(y∈Yc)such that ystrictly dominates (d◦f)(x)(strictly dominates f(x)). The corresponding image point is called weakly non-dominated. We can now begin to introduce statements about the relationship between particular solutions of Pand Pc. We first observe the connection between weakly non-dominated points of the two problems. Lemma 2.1 If yc,∗∈Ycis a weakly non-dominated point of Pcthen y∗:= d(yc,∗)∈ Yis weakly non-dominated for P. Proof For an arbitrary point yin the image Yof Pthere is at least one point yc∈Yc such that y=d(yc). Now consider the weakly non-dominated point yc,∗∈Yc. There must be at least one index i∈{1,...,k}such that yc,∗ i≤yc i, as otherwise ycwould strictly dominate yc,∗and yc,∗could not be a weakly nondominated point. For the same index iit follows by monotonicity of dithat y∗ i=di(yc,∗ i)≤di(yc i)=yi. This shows that ydoes not strictly dominate y∗.As ywas chosen arbitrary in Y,this shows that also y∗is a weakly non-dominated point.  Lemma 2.1 also implies that a non-dominated point is mapped to a weakly nondominated point. Unfortunately this point is not necessarily non-dominated anymore, as the following example shows: Example 2.2 Consider the following example (illustrated in Fig.1): •M={x∈R2|x2≤1} •k=2, f1(x)=x1,f2(x)=x2 •for t∈R:d1(t)=d2(t)and d1(t)=1ift≤−0.5, 2ift>−0.5. The point (−1,0)is a non-dominated point for problem Pc. However, by dit is mapped to the point (1,2). This point is dominated by (1,1), which is the image of the feasible solution (−0.5,−0.5)under d◦f. 123 Combining discrete and continuous information for… 157 Fig. 1 Illustration of Example 2.2. The image set Ycof the inner problem Pcis depicted in gray and its Pareto front in blue. The non-dominated point (−1,0), depicted in red, is mapped by dto the weakly non-dominated point (1,2).This point is dominated by (1,1) The example also shows that the reverse relationship does not hold. If d(yc) is a weakly non-dominated point, then the preimage can be strictly dominated. The point (−0.5,−0.5)is dominated in Pcby the point (−√2 2,−√2 2). However, d((−0.5,−0.5)) is a non-dominated point. This means that not all preimages of a non-dominated point are again (weakly) non-dominated. The following can be shown about the existence of non-dominated points in the preimage: Lemma 2.3 Let y∗∈Ybe a non-dominated point of P. Then there exists a nondominated point yc,∗∈Ycof Pcsuch that y∗=d(yc,∗). Proof Let Z=d−1(y∗)∩Yc denote the preimage of the non-dominated point y∗∈Ygiven in the statement. By assumption this set is nonempty. We want to show that we can find a point in Zthat is non-dominated for Pc. First consider any point z∈Zand a point z∈Ycsuch that: z i≤zifor all i∈{1,...,k}.(5) Then by monotonicity of diwe have: di(z i)≤di(zi)=y∗ ifor all i∈{1,...,k} As y∗is non-dominated, we must have: d(z)=y∗. 123 158 K. Teichert et al. This means that every point z∈Ycsatisfying the inequalities in Eq. (5) is already contained in Z. As we did not assume any continuity properties for d,thesetZis not necessarily closed. This is why we consider the closure of this set denoted by Z. As we assumed that fis continuous and Mis compact, the set Zis as a subset of Yc=f(M)compact. Now fix an arbitrary point z∗∈Zand consider the following optimization problem: min z∈Z k  i=1 zi s.t. zi≤z∗ ifor all i∈{1,...,k} The feasible set is compact and non-empty by construction. Moreover every feasible point of this problem is by the arguments given above in Z. This means that also an optimal solution yc,∗∈Ycto this problem is again in Z, which then means that d(yc,∗)=d(y∗). It remains to argue that yc,∗is instead a non-dominated point for Pc. Because of optimality, yc,∗cannot be dominated by any point in Z. Moreover, every point in Ycthat would dominate zalso satisfies (5) and is again in Z. Together this means that the constructed point cannot be dominated by any point in Ycand is indeed a non-dominated point for problem Pc. This Lemma is crucial for an algorithmic approach. It shows that we can find the Pareto front of the perhaps more complex problem Pby calculating points of the Pareto front of the continuous problem Pc. For completeness we show in the next example that the statement of Lemma 2.3 does not hold true, if one replaces non-dominated by weakly non-dominated. Example 2.4 Consider the same feasible set and the same objectives as in Example 2.2 but consider the following utility function: •for t∈R:d1(t)=d2(t)and d1(t)=1ift≤0.5, 2ift>0.5. For an illustration, see Fig.2. In this example, the point (2,1)is a weakly nondominated point for P. However, all points in Ycthat are mapped to this point are strictly dominated in Pc,e.g.by(−1,0). 3 Algorithms In this section, we discuss different algorithms tailored at calculating the Pareto front for a problem Pwith discrete utility mappings. That is, for the algorithms presented 123 Combining discrete and continuous information for… 159 Fig. 2 Illustration of Example 2.4. The image set Ycof the inner problem Pcis depicted in gray and its Pareto front in blue. Every point in the dark area is strictly dominated in the inner problem Pc, but it is the preimage of a weakly non-dominated point (1,2) of P Fig. 3 Illustration of exhaustively solving the multi-criteria problem P. The preimages of the points y∈ k i=1idivide the image space of the continuous problem Pcinto boxes. By finding specific solutions to the inner problem—which map to points on the continuous Pareto front depicted in blue—we can determine whether each box, and thus each y∈k i=1i, is non-dominated, dominated, or unattainable in this section to be applicable, we from now on explicitly require that Assumption 1.1 is satisfied. This means that for the image of Pwe have: Y⊆ k  i=1 i. The proposed algorithms utilize the special structure of P. On the one hand, all proposed methods employ a gradient-based solver to find either feasible or weakly efficient solutions xof Pby iteratively solving particular single-criteria optimization problems—so-called scalarizations—derived from the continuous inner problem Pc. On the other hand, the algorithms also take the discrete utility mappings into account, namely when choosing which scalarization to solve next and when evaluating their progress. 123 160 K. Teichert et al. The proposed algorithms aim to exhaustively solve the multi-criteria problem P (see Fig.3). This means that: •For all y∈k i=1i, the algorithm determines whether yis non-dominated, dominated or unattainable. •For any y∈k i=1ithat is non-dominated, the algorithm calculates a solution x∈Msuch that y=(d◦f)(x). In the applications we will introduce in Sect.4the cardinality of each individual i will not only be finite but also small (3–5). Additionally we will not have arbitrary many objectives in mind (in the examples 2–4). This means that it is possible to iterate over all elements in k i=1iand to save information for each element. However, our goal is to keep the computational effort – in the sense of optimization problems to solve over all elements – small. During application of the following algorithms a point y∈k i=1ican have four different current states encoded by σ(y): •σ(y)=attainable: The algorithm found an xwith (d◦f)(x)y. In this case we save xin X∗(y). •σ(y)=dominated: The algorithm determined a point ˜ y∈Ythat is attainable and that dominates y. •σ(y)=unattainable: The algorithm checked that there is no point xthat is mapped to yor any point that dominates y. •σ(y)=unknown: The algorithm did not make any statement about this point, yet. Note that we do not need to explicitly encode all points that are attainable but nondominated. If we found correctly all points that are attainable and of these points all that are dominated, then the complement of the dominated points are the non-dominated ones. We can also be sure to have calculated a preimage xfor each non-dominated y. As for each attainable point ywe found an xwith: (d◦f)(x)y. For a non-dominated point none of these inequalities can be strict and we must have: (d◦f)(x)=y. In the following, we call a point ya supported point of Pcwith weights w(see e.g. Ehrgott 2005) if it solves the following optimization problem: min y∈Yc k  i=1 wi·yi. 123 Combining discrete and continuous information for… 167 Fig. 6 The box checking algorithm, using the Pascoletti-Serafini scalarization, for a problem instance of 8 with k=2andK=4. In total, six iterations are necessary to exhaustively solve the problem d4 i(xi)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0.4if0≤xi≤0.4, 0.6if0.4<xi≤0.6, 0.8if0.6<xi≤0.8, 1.0ifxi>0.8 ∀i=1, .., k d5 i(xi)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0.4if0≤xi≤0.4, 0.5if0.4<xi≤0.5, 0.6if0.5<xi≤0.6, 0.8if0.6<xi≤0.8, 1.0ifxi>0.8 ∀i=1, .., k. The underlying continuous problem Pcof the randomized problems (8) is convex, allowing the application of all algorithms (A), (B) and (C). For a specific problem instance with k=2 and K=4, Fig.6illustrates the course of the box checking algorithm (A), and Fig.7depicts the course when using the combined algorithm for convex problems (C) instead. We see that for this instance, the combined algorithm for convex problems outperforms the box checking algorithm, finishing in 3 rather than 6 iterations. We can compare the performance of algorithms (A)–(C) by measuring the progress pover the iterations, with pbeing defined as: 123 168 K. Teichert et al. Fig. 7 The convex algorithm applied to a problem instance of Eq.8with k=2andK=4. Only three iterations are necessary to determine all non-dominated points. Note that the solution for one of the nondominated outcomes is obtained by convex combination of previously calculated points according to Lemma 3.3 (orange) p=1−|{y∈k i=1i|σ(y)unknown}| |k i=1i|.(9) Figure8shows the averaged progress for the artificial problem 8over varying number of objectives k∈{2,3,4}and image set cardinalities K∈{3,4,5}. To create each of the averaged plots, 10 randomly created problem instances were solved and the mean of obtained progress values pafter each iteration was calculated. The combined convex algorithm (C) performs best by exploiting the convexity of the problem. The next best choice is the combined algorithm that uses the supported solutions algorithm as a first phase. The box checking algorithm on its own does not perform as well. With higher dimension kand higher image set cardinality K,the differences in algorithm performance become more pronounced. 4.2 Radiotherapy planning example In radiotherapy planning, the aim is to deliver the prescribed dose to the target volumes while sparing nearby organs at risk. In intensity-modulated radiation therapy (IMRT), the irradiation—the so-called fluence—is delivered from different angles around the patient and can additionally be modulated over the cross section of each beam by moving collimator leaves in and out of the beam field. The fluence is represented by a vector x0 and constitutes the optimization variable of the multi-criteria radiotherapy planning problem (Küfer et al. 2002; Küfer et al. 2003; Craft et al. 2012). The dose influence matrix D depends on the patient anatomy as established by a CT scan and promotes the mapping from the fluence to the dose, such that the entries of Dxrepresent the dose for each voxel in the voxelized patient anatomy. The optimization objectives then evaluate the dose vector entries over the target volumes and organs at risk. These evaluation functions are continuous and often convex. Possible choices are minimum, maximum, mean, p-norms or one-sided p-norms. 123 Combining discrete and continuous information for… 169 Fig. 8 Averaged progress for algorithms (A)–(C): box checking algorithm (dark blue), combined algorithm with supported solutions algorithm as first phase (purple dotted) and combined algorithm with convex algorithm as first phase (red). The averages were taken over 10 randomized problem instances of 8.The number of objectives kwas varied from 2 to 4 and the image set cardinality Kwas varied from 3 to 5 Thus, the (continuous) multi-criteria radiotherapy planning problem has the properties required for the inner problem Pcin our setting. In clinical practice, certain threshold values for the objectives play a crucial role. In the case of organs at risk, violating a threshold is linked to specific side effects. For the targets, failing to meet a specific value may increase the probability of the tumor recurring. Often, these clinical goals are further categorized into those which represent the absolute minimum requirement, and others that correspond to an average or an ideal dose distribution in the volume. Taking these discretizations (“minimum”, “average”, “ideal”) into account, we obtain the structure of the continuous problem with discretized utility P. As an example, we consider the TG119 case. The anatomy of the case has a Ushaped target volume, an organ at risk (OAR) in the center of the target, and the surrounding normal tissue region. We define the beam geometry as 7 equidistantly spaces beams, see Fig. 9. To set up the optimization problem, we formulate one objective for each structure as detailed in Table 1. Additionally, for each objective we define the discretization mappings into the categories 3 (“minimum”), 2 (“average”), and 1 (“ideal”) by certain upper bounds, also given in Table 1, such that the discretized utility functions di 123 170 K. Teichert et al. Fig. 9 Geometry of the TG119 case. It has a U-shaped target volume (white), an organ at risk (magenta) in the center of the target, and surrounding normal tissue. Seven beams are spaced equidistantly (white lines) Table 1 Optimization model for the TG119 case: optimization objectives for target, OAR and tissue, and threshold values for the discretization into categories 1–3 Volume Objective Categories: 3 2 1 OAR 1 |OAR|v∈OAR(Dx)v25.0 23.0 20.0 Tissue 1 |tissue|v∈OAR(Dx)v8.0 6.0 5.0 Target v∈targetmax {0,60 −(Dx)v}22.0 1.5 1.0 (i=1,2,3)map to the smallest category for which fi(Dx)falls below the category’s upper bound. Again, we compared the algorithms (A)–(C). Figure10 shows the result obtained with all three algorithms. There are two Pareto-optimal solutions to the discretized problem: one where the target and the tissue dose quality are ideal while the OAR dose quality is average, and one where the OAR and tissue dose quality are ideal and the target dose quality is average. This is reflected in the dose volume histograms for the two solutions, where the first solution shows a better slope at the prescription dose level of 60 Gy, while the other solution exhibits a lower OAR curve. Figure11 shows the progress—as defined in (9)—for the different algorithms over the iterations. For this particular example, the algorithms do not differ much in their efficiency. The combined algorithm that uses the convex algorithm as a first phase performs the best, followed by the box checking algorithm and the other combined algorithm. 5 Conclusion In this paper we considered multi-criteria optimization problems which are based on a continuous problem. The outputs of the objectives are mapped using a discrete utility function making the overall multi-criteria optimization a discrete one. For example, such categories can be used to quickly find candidate solutions for a decision maker, especially when the Pareto front is no longer easily visualized. In a first step a decision maker might want to get a rough overview of the different alternatives instead of navigating locally. 123 Combining discrete and continuous information for… 171 Fig. 10 The result for the TG119 example. On the left side, the OAR mean objective and its discretization into categories is plotted against the tissue mean objective and its discretization for each of the 3 categories of the target underdose objective. There are two Pareto optimal outcomes in the discretized space. These are achieved by two efficient solutions xmapping to the marked points under the continuous objectives defined in Table 1(the target objective value being shown in purple.) The dose distributions for the two solutions—displayed as dose-volume histograms at the right—represent two distinct compromises, with the upper solution achieving a better target coverage and the lower solution a better sparing of the OAR Fig. 11 Progress for algorithms (A)–(C) for the TG119 radiotherapy planning example: box checking algorithm (dark blue), combined algorithm with supported solutions algorithm as first phase (purple), and combined algorithm with convex algorithm as first phase (red) 123 172 K. Teichert et al. In this paper, we first studied the connection between solutions of the underlying continuous problem and the combined multi-criteria problem. The results were used to introduce several algorithms that combine the discrete and continuous structure. In numerical examples, we were able to show that our approaches can save a lot of computation compared to a naive approach. For larger problems with more objectives and more categories, the experiments showed that it is beneficial to use as much information as possible and also to exploit convexity. Several aspects were beyond the scope of this paper, but may be of interest for future investigations. We focused on fully discrete utilities. In Sect. 2we already pointed out that this is not necessarily required from a theoretical point of view. All results hold for monotone functions in general. It would be interesting to look at combined continuous and discrete utilities. We also focused on problems with a moderate number of categories and objectives. This way it was not a problem to iterate over all possible category combinations. It would be interesting to study how the developed strategies transfer to this larger setting. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest The authors have no competing interests to declare that are relevant to the content of this article. No funding was received to assist with the preparation of this manuscript. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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