Original: Global Optimal K-Medoids Clustering of One Million Samples
Revised: Global Optimal K-Medoids Clustering of One Million Samples

Original: Abstract
Revised: Abstract

Original: We study the deterministic global optimization of the K-Medoids clustering problem.
Revised: We study the deterministic global optimization of the K-Medoids clustering problem.

Original: This work proposes a branch and bound (BB) scheme, in which a tailored
Revised: This work proposes a branch and bound (BB) scheme, in which a tailored

Original: Lagrangian relaxation method proposed in the 1970s is used to provide a lower bound at each BB node.
Revised: Lagrangian relaxation method proposed in the 1970s is used to provide a lower bound at each BB node.

Original: The lower bounding method already guarantees the maximum gap at the root node.
Revised: The lower bounding method already guarantees the maximum gap at the root node.

Original: A closed-form solution to the lower bound can be derived analytically without explicitly solving any optimization problems, and its computation can be easily parallelized.
Revised: A closed-form solution to the lower bound can be derived analytically without explicitly solving any optimization problems, and its computation can be easily parallelized.

Original: Moreover, with this lower bounding method, finite convergence to the global optimal solution can be guaranteed by branching only on the regions of medoids.
Revised: Moreover, with this lower bounding method, finite convergence to the global optimal solution can be guaranteed by branching only on the regions of medoids.

Original: We also present several tailored bound tightening techniques to reduce the search space and computational cost.
Revised: We also present several tailored bound tightening techniques to reduce the search space and computational cost.

Original: Extensive computational studies on 28 machine learning datasets demonstrate that our algorithm can provide a provable global optimal solution with an optimality gap of 0.1% within 4 hours on datasets with up to one million samples.
Revised: Extensive computational studies on 28 machine learning datasets demonstrate that our algorithm can provide a provable global optimal solution with an optimality gap of 0.1% within 4 hours on datasets with up to one million samples.

Original: Besides, our algorithm can obtain better or equal objective values than the heuristic method.
Revised: Besides, our algorithm can obtain better or equal objective values than the heuristic method.

Original: A theoretical proof of global convergence for our algorithm is also presented.
Revised: A theoretical proof of global convergence for our algorithm is also presented.

Original:  Introduction
Revised:  Introduction

Original: In this paper, we concentrate on the K-Medoids clustering problem: given a dataset, K-Medoids aims to select K samples from the dataset as cluster medoids that minimize the sum of dissimilarities from other samples to the closet medoids.
Revised: In this paper we concentrate on the K-Medoids clustering problem: given a dataset, K-Medoids aims to select K samples from the dataset as cluster medoids that minimize the sum of dissimilarities from other samples to the closet medoids.

Original: Here, selecting from existing samples as medoids is called the “medoids on samples” constraint.
Revised: Here, selecting from existing samples as medoids is called the “medoids on samples” constraint.

Original: K-Medoids is similar to the K-Means problem except for the arbitrary dissimilarity measurements and the “medoids on samples” constraint.
Revised: K-Medoids is similar to the K-Means problem except for the arbitrary dissimilarity measurements and the “medoids on samples” constraint.

Original: Hence, K-Medoids generally has better interpretability than K-Means [1].
Revised: Hence, K-Medoids generally has better interpretability than K-Means [1].

Original: K-Medoids problems are generally NP-hard to solve exactly.
Revised: K-Medoids problems are generally NP-hard to solve exactly.

Original: So there are many heuristic methods, such as PAM [2], CLARA [3], K-means-like method [4] and Fast-PAM [5].
Revised: So there are many heuristic methods, such as PAM [2], CLARA [3], K-means-like method [4] and Fast-PAM [5].

Original: However, none of them can guarantee reaching the global minimum.
Revised: However, none of them can guarantee reaching the global minimum.

Original: To solve the K-Medoids problem deterministically, back in the 1970s, Cornuejols et al.
Revised: To solve the K-Medoids problem deterministically, back in the 1970s, Cornuejols et al.

Original: [6] proposed a
Revised: [6] proposed a

Original: Lagrangian relaxation method for obtaining a lower bound of the optimal value, which can guarantee a relative optimal gap smaller than 1 /e .
Revised: Lagrangian relaxation method for obtaining a lower bound of the optimal value, which can guarantee a relative optimal gap smaller than 1 /e .

Original: Interestingly, to the best of our knowledge, this is the only application in which the Lagrangian duality gap can be guaranteed within a certain threshold.
Revised: Interestingly, to the best of our knowledge, this is the only application in which the Lagrangian duality gap can be guaranteed within a certain threshold.

Original: Several improvements to this method are proposed in the literature.
Revised: Several improvements to this method are proposed in the literature.

Original: One common direction is improving the heuristic process of Lagrangian dual problems.
Revised: One common direction is improving the heuristic process of Lagrangian dual problems.

Original: [7] constructed a semi-Lagrangian relaxation, then accelerated the heuristic process by a cutting plane method.
Revised: [7] constructed a semi-Lagrangian relaxation, then accelerated the heuristic process by a cutting plane method.

Original: It was further improved in [8] by fixing variables in the heuristic process.
Revised: It was further improved in [8] by fixing variables in the heuristic process.

Original: Their works can deal with medium-scale datasets (e.g.,samples).
Revised: Their works can deal with medium-scale datasets (e.g.,samples).

Original: Another common direction is column generation [9, 10].
Revised: Another common direction is column generation [9, 10].

Original: One limitation of these Lagrangian relaxation based methods is that there is no guarantee of convergence to the global optimal solution, due to the existence of duality gap.
Revised: One limitation of these Lagrangian relaxation based methods is that there is no guarantee of convergence to the global optimal solution, due to the existence of duality gap.

Original: To address the global optimum guarantee, a few works have utilized the branch and bound (BB) scheme in K-Medoids [11, 12, 13].
Revised: To address the global optimum guarantee, a few works have utilized the branch and bound (BB) scheme in K-Medoids [11, 12, 13].

Original: Christofides and Beasley [11] proposed a prototype by branching on binary variables and was capable of small-scale datasets (200 samples).
Revised: Christofides and Beasley [11] proposed a prototype by branching on binary variables and was capable of small-scale datasets (200 samples).

Original: This scheme was further improved by [12] through branching on semi-assignment constraints, which obtained solutions faster on small-scale datasets.
Revised: This scheme was further improved by [12] through branching on semi-assignment constraints, which obtained solutions faster on small-scale datasets.

Original: [13] presented a tighter form of the K-Medoids problem and was capable of medium-scale datasets (900 samples).
Revised: [13] presented a tighter form of the K-Medoids problem and was capable of medium-scale datasets (900 samples).

Original: Besides these specific works, several off-the-shelf solvers also contain well-implemented BB schemes, such as CPLEX [14] and Gurobi [15].
Revised: Besides these specific works, several off-the-shelf solvers also contain well-implemented BB schemes, such as CPLEX [14] and Gurobi [15].

Original: These solvers can typically address the K-Mediods problem with medium-scale datasets (e.g., 2000 samples).
Revised: These solvers can typically address the K-Mediods problem with medium-scale datasets (e.g., 2000 samples).

Original: The BB schemes used in the aforementioned works can only address medium-scale datasets because they need to branch on all the binary variables to guarantee convergence, which leads to poor scalability because the number of binary variables increases linearly as the number of samples.
Revised: The BB schemes used in the aforementioned works can only address medium-scale datasets because they need to branch on all the binary variables to guarantee convergence, which leads to poor scalability because the number of binary variables increases linearly as the number of samples.

Original: Therefore, we focus on the reduced-space BB scheme proposed in the stochastic programming community, which only needs to branch on the space of coupling variables (e.g., medoids).
Revised: Therefore, we focus on the reduced-space BB scheme proposed in the stochastic programming community, which only needs to branch on the space of coupling variables (e.g., medoids).

Original: This idea was first reported for stochastic mixed-integer linear programs in [16] and then extended to stochastic mixed-integer nonlinear programs in [17].
Revised: This idea was first reported for stochastic mixed-integer linear programs in [16] and then extended to stochastic mixed-integer nonlinear programs in [17].

Original: The convergence for stochastic nonlinear programs was proved in [18].
Revised: The convergence for stochastic nonlinear programs was proved in [18].

Original: Recently, the authors in [19] adopted this method and reported the capacity of solving the K-means clustering problem (a similar clustering task without the “medoids on samples” constraint) with 210,000 samples in parallel to an optimal gap of 2.5%.
Revised: Recently, the authors in [19] adopted this method and reported the capacity of solving the K-means clustering problem (a similar clustering task without the “medoids on samples” constraint) with 210,000 samples in parallel to an optimal gap of 2.5%.

Original: However, the convergence of the reduced-space BB scheme can only be guaranteed when it is combined with suitable lower bounding methods.
Revised: However, the convergence of the reduced-space BB scheme can only be guaranteed when it is combined with suitable lower bounding methods.

Original: In previous works, lower bounds are constructed by the Lagrangian relaxation [16] [17] [19] or removal [18] of the so-called non-anticipativity constraints (an implicit constraint that all the samples share the same coupling variables, that is Constraint 10 in this paper).
Revised: In previous works, lower bounds are constructed by the Lagrangian relaxation [16] [17] [19] or removal [18] of the so-called non-anticipativity constraints (an implicit constraint that all the samples share the same coupling variables, that is Constraint 10 in this paper).

Original: However, these methods are unsuitable for the K-Medoids problem, since they cannot address the "medoids on samples" constraint (a constraint that has a large dimension and connects all samples) and thus cannot ensure the convergence of the reduced-space BB scheme.
Revised: However, these methods are unsuitable for the K-Medoids problem, since they cannot address the "medoids on samples" constraint (a constraint that has a large dimension and connects all samples) and thus cannot ensure the convergence of the reduced-space BB scheme.

Original: Moreover, these lower bounding methods require the solution of many expensive mixed-integer nonlinear programs to global optimality at each node.
Revised: Moreover, these lower bounding methods require the solution of many expensive mixed-integer nonlinear programs to global optimality at each node.

Original: In contrast, the work of Cornuejols et al.
Revised: In contrast, the work of Cornuejols et al.

Original: [6] generates lower bounds for the K-Medoids problem by the Lagrangian relaxation of the semi-assignment constraint (Constraint 2b in this paper), and the formulation directly takes the "medoids on samples" constraint into consideration.
Revised: [6] generates lower bounds for the K-Medoids problem by the Lagrangian relaxation of the semi-assignment constraint (Constraint 2b in this paper), and the formulation directly takes the "medoids on samples" constraint into consideration.

Original: This method has a closed-form solution, and its computation can be easily parallelized.
Revised: This method has a closed-form solution, and its computation can be easily parallelized.

Original: Our Contributions: Firstly, we combine the method of reduced-space BB scheme and the work of Cornuejols et al.
Revised: Our Contributions: Firstly, we combine the method of reduced-space BB scheme and the work of Cornuejols et al.

Original: [6], and adapt these methods for the K-Medoids problem.
Revised: [6], and adapt these methods for the K-Medoids problem.

Original: The Lagrangian relaxation in [6] is used to generate a tight lower bound, which already guarantees a maximum gap at the root node.
Revised: The Lagrangian relaxation in [6] is used to generate a tight lower bound, which already guarantees a maximum gap at the root node.

Original: With this lower bound, we can prove the finite convergence (detailed proofs presented in the Appendix A) by branching only on the regions of medoids (the dimension of which is independent of the number of samples).
Revised: With this lower bound, we can prove the finite convergence (detailed proofs presented in the Appendix A) by branching only on the regions of medoids (the dimension of which is independent of the number of samples).

Original: Secondly, several tailored bound tightening techniques, including probing and feasibility based methods, are proposed to significantly reduce the search space and computational cost of each node.
Revised: Secondly, several tailored bound tightening techniques, including probing and feasibility based methods, are proposed to significantly reduce the search space and computational cost of each node.

Original: Lastly, we propose an upper bounding method by heuristic updates of the candidate solutions obtained in the lower bounding procedure.
Revised: Lastly, we propose an upper bounding method by heuristic updates of the candidate solutions obtained in the lower bounding procedure.

Original: These contributions enable our algorithm to be extremely scalable.
Revised: These contributions enable our algorithm to be extremely scalable.

Original: Capability For Million Scale Problems: we provide an open-source project in Julia and present numerical experiments on 28 datasets.
Revised: Capability For Million Scale Problems: we provide an open-source project in Julia and present numerical experiments on 28 datasets.

Original: Specifically, for two datasets with one million samples (which is 400 times larger than the state-of-art BB method), our algorithm can reach a small gap (0.1%) within one hour.
Revised: Specifically, for two datasets with one million samples (which is 400 times larger than the state-of-art BB method), our algorithm can reach a small gap (0.1%) within one hour.

Original: To the best of our knowledge, no other article has ever reported results on datasets of this scale for K-Medoids problems with a global convergence guarantee.
Revised: To the best of our knowledge, no other article has ever reported results on datasets of this scale for K-Medoids problems with a global convergence guarantee.

Original:  Preliminaries
Revised:  Preliminaries

Original: K-Medoids problem: we denote a dataset with S samples and A attributes as X = { x 1 , ..., x s , ..., x S } ∈ R S × A , in which x s =
Revised: K-Medoids problem: we denote a dataset with S samples and A attributes as X = { x 1 , ..., x s , ..., x S } ∈ R S × A , in which x s =

Original: [ x s, 1 , ..., x s,A ] ∈ R
Revised: [ x s, 1 , ..., x s,A ] ∈ R

Original: A is the s th sample, and x s,a is the a th attribute of s th sample.
Revised: A is the s th sample, and x s,a is the a th attribute of s th sample.

Original: The K-Medoids problem aims to select a subset from the dataset as the cluster medoids with the following objective:
Revised: The K-Medoids problem aims to select a subset from the dataset as the cluster medoids with the following objective:

Original: [Equation]
Revised: [Equation]

Original: where µ k :=
Revised: where µ k :=

Original: [ µ k 1 , ..., µ kA ] ∈ R
Revised: [ µ k 1 , ..., µ kA ] ∈ R

Original: A is the medoid of k th cluster, µ := { µ 1 , ..., µ K } is the medoids set,
Revised: A is the medoid of k th cluster, µ := { µ 1 , ..., µ K } is the medoids set,

Original: S :
Revised: S :

Original: = { 1 , ..., S } is the index set of samples in the dataset, K := { 1 , ..., K } is the index set of clusters,
Revised: = { 1 , ..., S } is the index set of samples in the dataset, K := { 1 , ..., K } is the index set of clusters,

Original: A := { 1 , ..., A } is the index set of attributes.
Revised: A := { 1 , ..., A } is the index set of attributes.

Original: We use µ ∈ X to represent the “medoids on samples” constraint in the K-Medoids problem.
Revised: We use µ ∈ X to represent the “medoids on samples” constraint in the K-Medoids problem.

Original: To facilitate the discussion of the BB scheme, we introduce ◦
Revised: To facilitate the discussion of the BB scheme, we introduce ◦

Original: M := { µ | ◦ µ ≤ µ ≤ ◦ ¯ µ } as the initial regions of medoids.
Revised: M := { µ | ◦ µ ≤ µ ≤ ◦ ¯ µ } as the initial regions of medoids.

Original: Here, ◦ µ and ◦ ¯ µ are the lower and upper bounds of medoids referred from data.
Revised: Here, ◦ µ and ◦ ¯ µ are the lower and upper bounds of medoids referred from data.

Original: Specifically, ◦ µ a = min s ∈ S x s,a and ◦ ¯ µ a = max s ∈ S x s,a are the bounds of the a th attribute.
Revised: Specifically, ◦ µ a = min s ∈ S x s,a and ◦ ¯ µ a = max s ∈ S x s,a are the bounds of the a th attribute.

Original: Note that the introduction of ◦ M does not affect the optimal solution.
Revised: Note that the introduction of ◦ M does not affect the optimal solution.

Original: Equivalently, the K-Medoids problem can also be expressed in an extensive form (EF) [20]:
Revised: Equivalently, the K-Medoids problem can also be expressed in an extensive form (EF) [20]:

Original: [Equation]
Revised: [Equation]

Original: where d s,j = || x s − x j || 22 is the distance between samples x s and x j ; y j is equal to 1 if sample x j is a medoid, and otherwise 0 ; b s,j is equal to 1 if sample x s belongs to the cluster whose medoid is x j , and otherwise 0 .
Revised: where d s,j = || x s − x j || 22 is the distance between samples x s and x j ; y j is equal to 1 if sample x j is a medoid, and otherwise 0 ; b s,j is equal to 1 if sample x s belongs to the cluster whose medoid is x j , and otherwise 0 .

Original: Note here d s,j is computed offline, so arbitrary dissimilarity measurements can be used without affecting the solution process.
Revised: Note here d s,j is computed offline, so arbitrary dissimilarity measurements can be used without affecting the solution process.

Original: Constraint 2b ensures that each sample x s must belong to one cluster.
Revised: Constraint 2b ensures that each sample x s must belong to one cluster.

Original: Constraint 2c guarantees that a total of K samples are selected as medoids.
Revised: Constraint 2c guarantees that a total of K samples are selected as medoids.

Original: Constraint 2d expresses that sample x s can be assigned to sample x j only if x j is the medoid of a cluster.
Revised: Constraint 2d expresses that sample x s can be assigned to sample x j only if x j is the medoid of a cluster.

Original: Lagrangian Relaxation of the K-Medoids problem: by the Lagrangian relaxation of the Constraint 2b, we can have a relaxed problem of the Extensive Form 2:
Revised: Lagrangian Relaxation of the K-Medoids problem: by the Lagrangian relaxation of the Constraint 2b, we can have a relaxed problem of the Extensive Form 2:

Original: [Equation]
Revised: [Equation]

Original: Constraints in the Extensive Form 2 except 2b ,
Revised: Constraints in the Extensive Form 2 except 2b ,

Original: [Equation]
Revised: [Equation]

Original: where λ s is the Lagrangian multiplier for s th sample.
Revised: where λ s is the Lagrangian multiplier for s th sample.

Original: We define the multiplier set as λ := { λ 1 , ..., λ S } .
Revised: We define the multiplier set as λ := { λ 1 , ..., λ S } .

Original: For a given λ , β LD ( ◦ M, λ ) can be solved analytically.
Revised: For a given λ , β LD ( ◦ M, λ ) can be solved analytically.

Original: Considering the objective function and constraints of the Relaxed Problem 3, we note that the optimal values for b s,j are given by
Revised: Considering the objective function and constraints of the Relaxed Problem 3, we note that the optimal values for b s,j are given by

Original: [Equation]
Revised: [Equation]

Original: We define the contribution of j th sample to the objective function as:
Revised: We define the contribution of j th sample to the objective function as:

Original: [Equation]
Revised: [Equation]

Original: The Lagrangian problem can be reduced to
Revised: The Lagrangian problem can be reduced to

Original: [Equation]
Revised: [Equation]

Original: Its closed-form solution can be easily derived by selecting y j corresponding to the K smallest ρ j .
Revised: Its closed-form solution can be easily derived by selecting y j corresponding to the K smallest ρ j .

Original: Lagrangian dual: a proper selection of λ may generate a tighter lower bound.
Revised: Lagrangian dual: a proper selection of λ may generate a tighter lower bound.

Original: Hence, we need to solve the Lagrangian dual problem to get the tightest lower bound:
Revised: Hence, we need to solve the Lagrangian dual problem to get the tightest lower bound:

Original: [Equation]
Revised: [Equation]

Original: There are several off-the-shelf heuristic methods to update λ , such as Sub-gradient method [11],
Revised: There are several off-the-shelf heuristic methods to update λ , such as Sub-gradient method [11],

Original: Volume method [21] and Cutting-Plane method [22].
Revised: Volume method [21] and Cutting-Plane method [22].

Original: The solution to the Dual Problem 7 provides a lower bound to the original problem, with β LD ( ◦ M ) ≤ z ( ◦ M ) .
Revised: The solution to the Dual Problem 7 provides a lower bound to the original problem, with β LD ( ◦ M ) ≤ z ( ◦ M ) .

Original: Particularly, for this problem, there is a unique guarantee of maximum gaps between the lower bounds and the global optimal value, which is
Revised: Particularly, for this problem, there is a unique guarantee of maximum gaps between the lower bounds and the global optimal value, which is

Original: [Equation]
Revised: [Equation]

Original: This guarantee is theoretically proved in [6, 23].
Revised: This guarantee is theoretically proved in [6, 23].

Original:  Lagrangian Based Lower Bound
Revised:  Lagrangian Based Lower Bound

Original: The aforementioned Lagrangian method cannot close the duality gap.
Revised: The aforementioned Lagrangian method cannot close the duality gap.

Original: To obtain a global optimal solution, we propose to combine it with the spatial branch and bound (BB) scheme, in which
Revised: To obtain a global optimal solution, we propose to combine it with the spatial branch and bound (BB) scheme, in which

Original: Lagrangian relaxation provides a lower bound for each BB node.
Revised: Lagrangian relaxation provides a lower bound for each BB node.

Original: Tailored Lagrangian Relaxation in BB nodes: Since our BB scheme branches on the regions of the medoids µ , at each BB node, we solve the problem with the partition set M ⊆ ◦ M .
Revised: Tailored Lagrangian Relaxation in BB nodes: Since our BB scheme branches on the regions of the medoids µ , at each BB node, we solve the problem with the partition set M ⊆ ◦ M .

Original: That is, a constraint µ ∈ M is added at each BB node.
Revised: That is, a constraint µ ∈ M is added at each BB node.

Original: We further define M k :=
Revised: We further define M k :=

Original: { µ k | µ k ≤ µ k ≤ ¯ µ k } as the medoid region of the k th cluster, where µ k and ¯ µ k are the lower and upper bounds of the k th medoids.
Revised: { µ k | µ k ≤ µ k ≤ ¯ µ k } as the medoid region of the k th cluster, where µ k and ¯ µ k are the lower and upper bounds of the k th medoids.

Original: Since the possible region of each cluster’s medoid can be different, we need to know the assignment of medoids to each cluster.
Revised: Since the possible region of each cluster’s medoid can be different, we need to know the assignment of medoids to each cluster.

Original: Hence, we replace y j with y kj in EF 2: y kj is equal to 1 if sample x j is selected as the medoid of the k th cluster, and otherwise 0 .
Revised: Hence, we replace y j with y kj in EF 2: y kj is equal to 1 if sample x j is selected as the medoid of the k th cluster, and otherwise 0 .

Original: The Constraint 2d is replaced by
Revised: The Constraint 2d is replaced by

Original: [Equation]
Revised: [Equation]

Original: Notice that µ k is not directly modelled in EF, we need to transform the medoid regions µ k ∈ M k to the constraints of y kj .
Revised: Notice that µ k is not directly modelled in EF, we need to transform the medoid regions µ k ∈ M k to the constraints of y kj .

Original: The regions of medoids implicitly control the selection of y kj because we have y kj = 0 , if x j ̸∈ M k .
Revised: The regions of medoids implicitly control the selection of y kj because we have y kj = 0 , if x j ̸∈ M k .

Original: Hence, we define the feasible index set of k th cluster’s medoid as
Revised: Hence, we define the feasible index set of k th cluster’s medoid as

Original: S k + ( M ) :
Revised: S k + ( M ) :

Original: = { j ∈ S | µ k ≤ x j ≤ ¯ µ k } .
Revised: = { j ∈ S | µ k ≤ x j ≤ ¯ µ k } .

Original: In this way, we can reformulate the reduced Lagrangian
Revised: In this way, we can reformulate the reduced Lagrangian

Original: Problem 6 at the BB node with M as follow:
Revised: Problem 6 at the BB node with M as follow:

Original: [Equation]
Revised: [Equation]

Original: with the Lagrangian dual problem defined as β LD ( M ) = max λ β LD ( M, λ ) .
Revised: with the Lagrangian dual problem defined as β LD ( M ) = max λ β LD ( M, λ ) .

Original: The optimal value of y kj in Problem 8 can be obtained by the following procedure: (1) initialize k = 

Revised: The optimal value of y kj in Problem 8 can be obtained by the following procedure: (1) initialize k =

Original: and J ∗ = ∅ (2) let j k ∗ be the index of smallest ρ j ( λ ) subject to j ∈ S k
Revised: and J ∗ = ∅ (2) let j k ∗ be the index of smallest ρ j ( λ ) subject to j ∈ S k

Original: + ( M ) and j / ∈ J ∗ ; (3) set y kj =
Revised: + ( M ) and j / ∈ J ∗ ; (3) set y kj =

Original: if j = j k ∗ , otherwise, 0; (4) add j k ∗ to J ∗ ; (5) update k = k + 1 and return to (2) until k = K .
Revised: if j = j k ∗ , otherwise, 0; (4) add j k ∗ to J ∗ ; (5) update k = k + 1 and return to (2) until k = K .

Original: Parallel implementation: we implement a parallel version of the tailored Lagrangian relaxation, in which the computations of contributions, ρ j ( λ ) , are performed in each process.
Revised: Parallel implementation: we implement a parallel version of the tailored Lagrangian relaxation, in which the computations of contributions, ρ j ( λ ) , are performed in each process.

Original: At the beginning of the BB procedure, given P processes, the index set S of the whole dataset is evenly divided into P subsets, S p , p = 1 , .., P .
Revised: At the beginning of the BB procedure, given P processes, the index set S of the whole dataset is evenly divided into P subsets, S p , p = 1 , .., P .

Original: Then, each process computes and stores the distance assigned to it, which is d s,j , s ∈ S , j ∈ S p .
Revised: Then, each process computes and stores the distance assigned to it, which is d s,j , s ∈ S , j ∈ S p .

Original: At each BB node with M , denote the feasible index set of all the medoids as
Revised: At each BB node with M , denote the feasible index set of all the medoids as

Original: [Equation]
Revised: [Equation]

Original: Then, each process computes the assigned contributions using the stored d s,j , which is ρ j ( λ ) , ∀ j ∈
Revised: Then, each process computes the assigned contributions using the stored d s,j , which is ρ j ( λ ) , ∀ j ∈

Original: S p ∩ S + ( M ) .
Revised: S p ∩ S + ( M ) .

Original: The remaining parts of the parallel implementation (e.g., update of λ ) are identical to the serial version.
Revised: The remaining parts of the parallel implementation (e.g., update of λ ) are identical to the serial version.

Original:  Tailored Reduced-space Branch and Bound Scheme
Revised:  Tailored Reduced-space Branch and Bound Scheme

Original: This section introduces the tailored reduced-space branch and bound algorithm for the K-Medoids problem, which is detailed in Algorithm 1.
Revised: This section introduces the tailored reduced-space branch and bound algorithm for the K-Medoids problem, which is detailed in Algorithm 1.

Original: Specifically, starting from the root node with ◦ M , the BB scheme recursively partitions the medoid regions.
Revised: Specifically, starting from the root node with ◦ M , the BB scheme recursively partitions the medoid regions.

Original: At each iteration, the algorithm first selects the node with the lowest lower bound and denotes it as M .
Revised: At each iteration, the algorithm first selects the node with the lowest lower bound and denotes it as M .

Original: Then, bound tightening techniques are applied to get a tightened medoid region ˆ M .
Revised: Then, bound tightening techniques are applied to get a tightened medoid region ˆ M .

Original: The lower and upper bounds of the tightened node are computed and updated sequentially.
Revised: The lower and upper bounds of the tightened node are computed and updated sequentially.

Original: If the gap is larger than the tolerance, we branch on the medoid variable µ k a with the maximum range to get two sub-nodes M 1 and M 2 with relint ( M 1 ) ∩ relint ( M 2 ) = ∅ , where relint ( M ) is the relative interior of M .
Revised: If the gap is larger than the tolerance, we branch on the medoid variable µ k a with the maximum range to get two sub-nodes M 1 and M 2 with relint ( M 1 ) ∩ relint ( M 2 ) = ∅ , where relint ( M ) is the relative interior of M .

Original: On the convergence of the algorithm, we have the following theorem with detailed proofs in Appendix A:
Revised: On the convergence of the algorithm, we have the following theorem with detailed proofs in Appendix A:

Original: Theorem 1.
Revised: Theorem 1.

Original: Algorithm 1 is convergent to the global optimal solution after a finite step L , with β L =
Revised: Algorithm 1 is convergent to the global optimal solution after a finite step L , with β L =

Original: z = α L , by only branching on the space of µ .
Revised: z = α L , by only branching on the space of µ .

Original: ∈ X ∩
Revised: ∈ X ∩

Original: , do 2: if x s is not assigned in the parent node then 3: if d
Revised: , do 2: if x s is not assigned in the parent node then 3: if d

Original: −
Revised: −

Original: ,min , ∀ k − ∈ K \ k +
Revised: ,min , ∀ k − ∈ K \ k +

Original: Assign x s to k + th cluster.
Revised: Assign x s to k + th cluster.

Original: 5: end if 6: end if 7: end for
Revised: 5: end if 6: end if 7: end for

Original: 1: for Cluster k ∈ K do 2:
Revised: 1: for Cluster k ∈ K do 2:

Original: Obtain the assigned set S kA according to Alg.3.
Revised: Obtain the assigned set S kA according to Alg.3.

Original: Compute RHS =
Revised: Compute RHS =

Original: α
Revised: α

Original: − β
Revised: − β

Original: M )
Revised: M )

Original: 4: for Attribute
Revised: 4: for Attribute

Original: a ∈
Revised: a ∈

Original: A do 5:
Revised: A do 5:

Original: Obtain the tightened node, µ ka ≤ µ ka } , by solving the inequation 13.
Revised: Obtain the tightened node, µ ka ≤ µ ka } , by solving the inequation 13.

Original: Update
Revised: Update

Original: 7: end for 8: end for
Revised: 7: end for 8: end for

Original: 4.1 Lower Bounds
Revised: 4.1 Lower Bounds

Original: Basic lower bound:
Revised: Basic lower bound:

Original: Besides the Lagrangian based lower bound, we also introduce a basic lower bound β basic .
Revised: Besides the Lagrangian based lower bound, we also introduce a basic lower bound β basic .

Original: Although it is not as tight as the Lagrangian based lower bound β LD , its computational cost is significantly lower.
Revised: Although it is not as tight as the Lagrangian based lower bound β LD , its computational cost is significantly lower.

Original: Thus it is useful when the BB tree is deep with small medoid regions.
Revised: Thus it is useful when the BB tree is deep with small medoid regions.

Original: For
Revised: For

Original: K-Medoids 1, there is an implicit constraint that all the samples share the same medoid set:
Revised: K-Medoids 1, there is an implicit constraint that all the samples share the same medoid set:

Original: [Equation]
Revised: [Equation]

Original: (10) in which µ ks is the k th medoid for the s th sample.
Revised: (10) in which µ ks is the k th medoid for the s th sample.

Original: By removing the “medoids on samples” constraint µ ∈ X and Constraint 10, we have the basic lower bound β basic ( M ) with β basic ( M ) ≤ z ( M ) :
Revised: By removing the “medoids on samples” constraint µ ∈ X and Constraint 10, we have the basic lower bound β basic ( M ) with β basic ( M ) ≤ z ( M ) :

Original: [Equation]
Revised: [Equation]

Original: We define d ks,min ( M ) :
Revised: We define d ks,min ( M ) :

Original: = min µ k ∈ M k
Revised: = min µ k ∈ M k

Original: || x s − µ k || 22 and obviously β basic ( M )
Revised: || x s − µ k || 22 and obviously β basic ( M )

Original: = (cid:80) s
Revised: = (cid:80) s

Original: ∈ S min
Revised: ∈ S min

Original: k ∈K d
Revised: k ∈K d

Original: ks,min ( M ) .
Revised: ks,min ( M ) .

Original: The analytic solution to d ks,min ( M ) can be easily derived: µ ks,a = mid { µ ka , x s,a , ¯ µ ka } , a ∈ A , in which µ ka and ¯ µ ka are the bounds of the a th attribute in k th medoid region.
Revised: The analytic solution to d ks,min ( M ) can be easily derived: µ ks,a = mid { µ ka , x s,a , ¯ µ ka } , a ∈ A , in which µ ka and ¯ µ ka are the bounds of the a th attribute in k th medoid region.

Original: Lagrangian based lower bound: as shown in Algorithm 5, by combining the basic and LD lower bound methods, we give the final form of lower bounds: β ( M )
Revised: Lagrangian based lower bound: as shown in Algorithm 5, by combining the basic and LD lower bound methods, we give the final form of lower bounds: β ( M )

Original: =
Revised: =

Original: max { β basic ( M ) , β LD ( M ) } .
Revised: max { β basic ( M ) , β LD ( M ) } .

Original: We note that because β basic ( M ) ≤ z ( M ) and β LD ( M ) ≤ z ( M ) , we have β ( M ) ≤ z ( M ) .
Revised: We note that because β basic ( M ) ≤ z ( M ) and β LD ( M ) ≤ z ( M ) , we have β ( M ) ≤ z ( M ) .

Original: Besides, according to the guarantee of maximum gap between the Lagrangian based lower bound and the global optimal value, we can also derive a guarantee of maximum gaps for β ( M ) :
Revised: Besides, according to the guarantee of maximum gap between the Lagrangian based lower bound and the global optimal value, we can also derive a guarantee of maximum gaps for β ( M ) :

Original: Proposition 2.
Revised: Proposition 2.

Original: [Equation]
Revised: [Equation]

Original: where z
Revised: where z

Original: [Equation]
Revised: [Equation]

Original: 4.2 Upper Bounds
Revised: 4.2 Upper Bounds

Original: Upper bounds of the K-Medoids problem can be obtained by selecting a feasible set of medoids ˜ µ ∈ M ∩ X .
Revised: Upper bounds of the K-Medoids problem can be obtained by selecting a feasible set of medoids ˜ µ ∈ M ∩ X .

Original: Given a feasible solution, we can obtain the assignments of samples and object value through Equation 1, without solving any optimization problem explicitly.
Revised: Given a feasible solution, we can obtain the assignments of samples and object value through Equation 1, without solving any optimization problem explicitly.

Original: However, the selection of ˜ µ will critically influence the efficiency of convergence.
Revised: However, the selection of ˜ µ will critically influence the efficiency of convergence.

Original: Hence, in our implementation, we introduce two efficient upper bounding methods:
Revised: Hence, in our implementation, we introduce two efficient upper bounding methods:

Original: Root node:
Revised: Root node:

Original: we use
Revised: we use

Original: the K-Means-
Revised: the K-Means-

Original: like
Revised: like

Original: method
Revised: method

Original: [4] as a backbone.
Revised: [4] as a backbone.

Original: Because the K-Means-like method is sensitive
Revised: Because the K-Means-like method is sensitive

Original: to the initial seeds
Revised: to the initial seeds

Original: and apt to the local minimum [5], we run this method several times with
Revised: and apt to the local minimum [5], we run this method several times with

Original: random
Revised: random

Original: seeds to get a candidate solution set.
Revised: seeds to get a candidate solution set.

Original: Then, we improve the quality of the upper
Revised: Then, we improve the quality of the upper

Original: bound
Revised: bound

Original: by feeding the
Revised: by feeding the

Original: candidate
Revised: candidate

Original: solution set as initial
Revised: solution set as initial

Original: guesses
Revised: guesses

Original: to Evolutionary Centers Algorithm [24].
Revised: to Evolutionary Centers Algorithm [24].

Original: Child nodes: we utilize the candidate solution obtained in the lower bounding method to update the upper bounds.
Revised: Child nodes: we utilize the candidate solution obtained in the lower bounding method to update the upper bounds.

Original: Specifically, we first compute the corresponding assignments, then feed these assignments as the initial solution to the K-Means-like method.
Revised: Specifically, we first compute the corresponding assignments, then feed these assignments as the initial solution to the K-Means-like method.

Original: It should be noted that this K-Meanslike method [4] gives a solution full-filling the "medoids on samples" constraint in K-Medoids.
Revised: It should be noted that this K-Meanslike method [4] gives a solution full-filling the "medoids on samples" constraint in K-Medoids.

Original:  Bound Tightening Techniques
Revised:  Bound Tightening Techniques

Original: Here, we introduce some bound tightening techniques tailored for the K-Medoids problem to reduce the search regions M at each BB node while guaranteeing the optimal solution is not excluded.
Revised: Here, we introduce some bound tightening techniques tailored for the K-Medoids problem to reduce the search regions M at each BB node while guaranteeing the optimal solution is not excluded.

Original: These bound tightening techniques are deployed before lower and upper bounds calculation.
Revised: These bound tightening techniques are deployed before lower and upper bounds calculation.

Original: 5.1 Probing
Revised: 5.1 Probing

Original: Probing is a technique to reduce bound regions by exploiting the inequalities in a mixed integer programming problem [25].
Revised: Probing is a technique to reduce bound regions by exploiting the inequalities in a mixed integer programming problem [25].

Original: This paper exploits the implicit inequality that the lower bound of a region containing the optimal solution is smaller than the current best upper bound.
Revised: This paper exploits the implicit inequality that the lower bound of a region containing the optimal solution is smaller than the current best upper bound.

Original: As shown in Algorithm 2, it tentatively restricts the bounds of µ to a subinterval and then computes the corresponding lower bound.
Revised: As shown in Algorithm 2, it tentatively restricts the bounds of µ to a subinterval and then computes the corresponding lower bound.

Original: If the lower bound is larger than the best upper bound, we can conclude that no optimal solution exists within the subinterval, and tighten the bound on the variable accordingly.
Revised: If the lower bound is larger than the best upper bound, we can conclude that no optimal solution exists within the subinterval, and tighten the bound on the variable accordingly.

Original: The computation costs are trivial since we use the basic lower bounding method.
Revised: The computation costs are trivial since we use the basic lower bounding method.

Original: 5.2 Feasibility Based Bound Tightening
Revised: 5.2 Feasibility Based Bound Tightening

Original: This subsection introduces the Feasibility Based Bound Tightening (FBBT) technique.
Revised: This subsection introduces the Feasibility Based Bound Tightening (FBBT) technique.

Original: It is based on the observation that the assignment of some samples to clusters can be predetermined.
Revised: It is based on the observation that the assignment of some samples to clusters can be predetermined.

Original: [Figure]
Revised: [Figure]

Original: Cluster assignment: if we have || x s − µ k
Revised: Cluster assignment: if we have || x s − µ k

Original: + ||
Revised: + ||

Original: 22 <
Revised: 22 <

Original: || x s − µ k − || 22 ,
Revised: || x s − µ k − || 22 ,

Original: ∀ k − ∈ K \ k + , then obviously sample x s is in the k + th cluster.
Revised: ∀ k − ∈ K \ k + , then obviously sample x s is in the k + th cluster.

Original: However, the value of µ here is not known before solving the overall problem.
Revised: However, the value of µ here is not known before solving the overall problem.

Original: Recall that we have d ks,min ( M ):= min µ k ∈ M k
Revised: Recall that we have d ks,min ( M ):= min µ k ∈ M k

Original: || x s − µ k || 22 .
Revised: || x s − µ k || 22 .

Original: Similarly, we also define d ks,max ( M ):= max µ k ∈ M k
Revised: Similarly, we also define d ks,max ( M ):= max µ k ∈ M k

Original: || x s − µ k || 22 as the maximum distance between x s and µ k with µ k ∈ M k .
Revised: || x s − µ k || 22 as the maximum distance between x s and µ k with µ k ∈ M k .

Original: At a BB node with partition M , if we have d k + s,max ( M ) <
Revised: At a BB node with partition M , if we have d k + s,max ( M ) <

Original: d k − s,min ( M ) , ∀ k − ∈ K \ k + , then sample x s is guaranteed to be assigned to the k + th cluster.
Revised: d k − s,min ( M ) , ∀ k − ∈ K \ k + , then sample x s is guaranteed to be assigned to the k + th cluster.

Original: Based on this observation, we can check whether the cluster of some samples can be determined in a BB node as shown in Algorithm 3.
Revised: Based on this observation, we can check whether the cluster of some samples can be determined in a BB node as shown in Algorithm 3.

Original: Recall that d
Revised: Recall that d

Original: ks,min ( M ) is already computed as a byproduct of the basic lower bound in Section 3.
Revised: ks,min ( M ) is already computed as a byproduct of the basic lower bound in Section 3.

Original: Similarly, since the cluster regions for each node are rectangles, d ks,max ( M ) can also be analytically computed.
Revised: Similarly, since the cluster regions for each node are rectangles, d ks,max ( M ) can also be analytically computed.

Original: Besides, since child nodes are subsets of the parent node, once a sample is assigned to a cluster in a parent node, it is also assigned to the same cluster in its child nodes.
Revised: Besides, since child nodes are subsets of the parent node, once a sample is assigned to a cluster in a parent node, it is also assigned to the same cluster in its child nodes.

Original: Figure 1 shows a simple illustration of this cluster assignment scheme.
Revised: Figure 1 shows a simple illustration of this cluster assignment scheme.

Original: FBBT:
Revised: FBBT:

Original: In a node with M , we denote the index sets of samples assigned to cluster k as S kA .
Revised: In a node with M , we denote the index sets of samples assigned to cluster k as S kA .

Original: FBBT infers the regions of µ k from the following inequation:
Revised: FBBT infers the regions of µ k from the following inequation:

Original: [Equation]
Revised: [Equation]

Original: where α l is the current best upper bound, β ( M ) − represents the lower bound contributions of samples unassigned to cluster k (either samples assigned to other clusters or samples whose assignments are undetermined).
Revised: where α l is the current best upper bound, β ( M ) − represents the lower bound contributions of samples unassigned to cluster k (either samples assigned to other clusters or samples whose assignments are undetermined).

Original: Similar to the calculation of the basic lower bounding method, we have β ( M ) − =
Revised: Similar to the calculation of the basic lower bounding method, we have β ( M ) − =

Original: (cid:80)
Revised: (cid:80)

Original: [Equation]
Revised: [Equation]

Original: As shown in Algorithm 4, FBBT is applied for each cluster.
Revised: As shown in Algorithm 4, FBBT is applied for each cluster.

Original: Specifically, for the k th medoid region, 1) obtain S kA according to the results of Algorithm 3; 2) compute the right-hand-side, RHS =
Revised: Specifically, for the k th medoid region, 1) obtain S kA according to the results of Algorithm 3; 2) compute the right-hand-side, RHS =

Original: α l − β ( M ) − ; 3) for each attribute a ∈ A , obtain the tightened region, ˆ M ka =
Revised: α l − β ( M ) − ; 3) for each attribute a ∈ A , obtain the tightened region, ˆ M ka =

Original: { µ ka | ˆ µ ka ≤ µ ka ≤ ˆ¯ µ ka } , by solving the following inequation:
Revised: { µ ka | ˆ µ ka ≤ µ ka ≤ ˆ¯ µ ka } , by solving the following inequation:

Original: [Equation]
Revised: [Equation]

Original: Since 13 is a simple quadratic inequation, we can analytically solve the problem to obtain the lower and upper bound of µ ka as ˆ µ ka and ˆ¯ µ ka .
Revised: Since 13 is a simple quadratic inequation, we can analytically solve the problem to obtain the lower and upper bound of µ ka as ˆ µ ka and ˆ¯ µ ka .

Original: Finally, update ˆ M ka by ˆ M ka ← ˆ M ka ∩ M ka .
Revised: Finally, update ˆ M ka by ˆ M ka ← ˆ M ka ∩ M ka .

Original: 5.3 Miscellaneous
Revised: 5.3 Miscellaneous

Original: Symmetric-breaking: we enforce a symmetric-breaking constraint on the first attribute, i.e., µ k 1 ≤ µ k + , ∀ k
Revised: Symmetric-breaking: we enforce a symmetric-breaking constraint on the first attribute, i.e., µ k 1 ≤ µ k + , ∀ k

Original: = 1 , ..., K − 1 .
Revised: = 1 , ..., K − 1 .

Original: It is used to tighten the regions of M at each BB node.
Revised: It is used to tighten the regions of M at each BB node.

Original: Bound tightening according to the “medoids on samples” constraint: since the K-Medoids problem requires medoids to be selected from samples in the dataset, we can further tighten the regions of M by ˆ M = { µ | ˆ µ ≤ µ ≤ ˆ¯ µ } , ˆ µ ka = min s ∈ M ∩ X x s,a , and ˆ¯ µ ka = max s ∈ M ∩ X x s,a .
Revised: Bound tightening according to the “medoids on samples” constraint: since the K-Medoids problem requires medoids to be selected from samples in the dataset, we can further tighten the regions of M by ˆ M = { µ | ˆ µ ≤ µ ≤ ˆ¯ µ } , ˆ µ ka = min s ∈ M ∩ X x s,a , and ˆ¯ µ ka = max s ∈ M ∩ X x s,a .

Original: 5.4 Effects on Computation
Revised: 5.4 Effects on Computation

Original: With a reduced search space, these bound tightening techniques can significantly reduce the number of
Revised: With a reduced search space, these bound tightening techniques can significantly reduce the number of

Original: BB nodes to be explored.
Revised: BB nodes to be explored.

Original: Moreover, they can also accelerate the calculation of bounds.
Revised: Moreover, they can also accelerate the calculation of bounds.

Original: For example, for the Lagrangian Problem 8, we need to calculate the contributions, ρ j , of samples belonging to the overall feasible set of medoids, ˆ S + = ˆ S 1+ ∪ ˆ S 2+ ∪ ... ∪ ˆ S K + .
Revised: For example, for the Lagrangian Problem 8, we need to calculate the contributions, ρ j , of samples belonging to the overall feasible set of medoids, ˆ S + = ˆ S 1+ ∪ ˆ S 2+ ∪ ... ∪ ˆ S K + .

Original: A reduced search space ˆ M means a reduced overall feasible selection set, which further leads to fewer calculations and faster speeds.
Revised: A reduced search space ˆ M means a reduced overall feasible selection set, which further leads to fewer calculations and faster speeds.

Original:  Experiments and Discussions
Revised:  Experiments and Discussions

Original: To evaluate the performance, we use 25 datasets from the UCI Machine Learning Repository [26], one dataset called PR2392 from [27], one dataset called HEMI[28] and one synthetic dataset from [19].
Revised: To evaluate the performance, we use 25 datasets from the UCI Machine Learning Repository [26], one dataset called PR2392 from [27], one dataset called HEMI[28] and one synthetic dataset from [19].

Original: The datasets from UCI Repository are in the clustering category with multivariate numerical attributes and range from 100 to 2,458,285 instances.
Revised: The datasets from UCI Repository are in the clustering category with multivariate numerical attributes and range from 100 to 2,458,285 instances.

Original: We implement our branch and bound KMedoids algorithm with Lagrangian based lower bound method (BB+LD) in Julia 1.6.1.
Revised: We implement our branch and bound KMedoids algorithm with Lagrangian based lower bound method (BB+LD) in Julia 1.6.1.

Original: We also implement four other methods to compare the performance, including the extensive form of KMedoids problem solved by the state-of-art global optimizer CPLEX 20.1.0 [14] (CPLEX), the heuristic method in the root node (Heuristic), the pure Lagrangian relaxation method without branch and bound (LD), and the state-of-art BB method for K-Means in [19] adapting to K-Medoids by adding "medoids on samples" constraint (BB+Basic).
Revised: We also implement four other methods to compare the performance, including the extensive form of KMedoids problem solved by the state-of-art global optimizer CPLEX 20.1.0 [14] (CPLEX), the heuristic method in the root node (Heuristic), the pure Lagrangian relaxation method without branch and bound (LD), and the state-of-art BB method for K-Means in [19] adapting to K-Medoids by adding "medoids on samples" constraint (BB+Basic).

Original: We executed all the experiments on a highperformance computing cluster, of which each node contains 40 Intel cores at 2.4 GHz and
Revised: We executed all the experiments on a highperformance computing cluster, of which each node contains 40 Intel cores at 2.4 GHz and

Original: GB RAM.
Revised: GB RAM.

Original: We present the results of cluster number K = 3 in the main body with additional results K = 5 and K = 10 in Appendix C.
Revised: We present the results of cluster number K = 3 in the main body with additional results K = 5 and K = 10 in Appendix C.

Original: The complete code files can be found in https: //github.com/
Revised: The complete code files can be found in https: //github.com/

Original: YankaiGroup/global_kmedoids_clustering
Revised: YankaiGroup/global_kmedoids_clustering

Original: The numerical performances are compared using three indicators, including upper bound value (UB), relative optimal gap, and the number of solved nodes.
Revised: The numerical performances are compared using three indicators, including upper bound value (UB), relative optimal gap, and the number of solved nodes.

Original: Here, UB is the best upper bound in all the iterations.
Revised: Here, UB is the best upper bound in all the iterations.

Original: The relative optimal gap is the relative difference between the best lower and upper bounds, which is calculated by Gap (%) :=
Revised: The relative optimal gap is the relative difference between the best lower and upper bounds, which is calculated by Gap (%) :=

Original: [Equation]
Revised: [Equation]

Original: This relative gap represents the worst gap of the found solution from the global optimal solution.
Revised: This relative gap represents the worst gap of the found solution from the global optimal solution.

Original: Notably, this gap is a particular property of deterministic global optimization algorithms.
Revised: Notably, this gap is a particular property of deterministic global optimization algorithms.

Original: Heuristic methods can not quantify the worst gap from the global optimal solution.
Revised: Heuristic methods can not quantify the worst gap from the global optimal solution.

Original: The number of solved nodes represents the total number of BB nodes.
Revised: The number of solved nodes represents the total number of BB nodes.

Original: All the deterministic methods CPLEX, LD, BB+Basics, and BB+LD, share the same terminal criteria, including (1) Gap ≤ 0 .
Revised: All the deterministic methods CPLEX, LD, BB+Basics, and BB+LD, share the same terminal criteria, including (1) Gap ≤ 0 .

Original: 1% , (2) the solving time reaches 4 hours, (3) the number of solved nodes reaches 5 million.
Revised: 1% , (2) the solving time reaches 4 hours, (3) the number of solved nodes reaches 5 million.

Original: Both LD and BB+LD use the same sub-gradient method called the Volume method [21].
Revised: Both LD and BB+LD use the same sub-gradient method called the Volume method [21].

Original: Particularly, for BB+LD, the maximum iteration of Lagrangian multiplier updates in each node is 10.
Revised: Particularly, for BB+LD, the maximum iteration of Lagrangian multiplier updates in each node is 10.

Original: For LD, the maximum iteration is 10,000.
Revised: For LD, the maximum iteration is 10,000.

Original: In the heuristic method, we execute the K-Means-like method 20 times with a fixed seed set including 20 different seeds, then feed the solution set as initial guesses to ECA to get the best solutions.
Revised: In the heuristic method, we execute the K-Means-like method 20 times with a fixed seed set including 20 different seeds, then feed the solution set as initial guesses to ECA to get the best solutions.

Original: [Table]
Revised: [Table]

Original: 6.1 Serial Results
Revised: 6.1 Serial Results

Original: Table 1 shows the serial numerical results for the datasets with less than 100,000 samples.
Revised: Table 1 shows the serial numerical results for the datasets with less than 100,000 samples.

Original: In terms of best found upper bound (UB), the deterministic global optimization methods (CPLEX, BB+Basic, and BB+LD) can obtain better UB than the heuristic method in several datasets (e.g., IRIS, BM,
Revised: In terms of best found upper bound (UB), the deterministic global optimization methods (CPLEX, BB+Basic, and BB+LD) can obtain better UB than the heuristic method in several datasets (e.g., IRIS, BM,

Original: RDS).
Revised: RDS).

Original: While we try to obtain the best heuristic results by multiple initializations, there is still no guarantee for the heuristic method to reach the global optimum.
Revised: While we try to obtain the best heuristic results by multiple initializations, there is still no guarantee for the heuristic method to reach the global optimum.

Original: Moreover, the heuristic method can not provide an optimal gap to estimate the solution quality.
Revised: Moreover, the heuristic method can not provide an optimal gap to estimate the solution quality.

Original: Notably, our method and CPLEX provide the same upper bounds and gaps for the datasets smaller than PR2392 (2392 samples).
Revised: Notably, our method and CPLEX provide the same upper bounds and gaps for the datasets smaller than PR2392 (2392 samples).

Original: As for the gaps and running times, CPLEX and LD can obtain a small gap ( ≤ 0.1%) with less time when datasets are smaller than HCV (572 samples).
Revised: As for the gaps and running times, CPLEX and LD can obtain a small gap ( ≤ 0.1%) with less time when datasets are smaller than HCV (572 samples).

Original: However, when datasets are larger than HCV, our method, BB+LD, can reach the small gap with much less time than CPLEX and LD.
Revised: However, when datasets are larger than HCV, our method, BB+LD, can reach the small gap with much less time than CPLEX and LD.

Original: Remarkably, our method can reach the small gap in all the datasets up to 100,000 samples within 4 hours.
Revised: Remarkably, our method can reach the small gap in all the datasets up to 100,000 samples within 4 hours.

Original: CPLEX fails to give a solution because of the out-of-memory error in the root node when datasets are larger than PR2392 (2392 samples).
Revised: CPLEX fails to give a solution because of the out-of-memory error in the root node when datasets are larger than PR2392 (2392 samples).

Original: LD and BB+Basic methods may not reach the small gap within 4 hours or maximum iteration limits when datasets are larger than BM (249 samples).
Revised: LD and BB+Basic methods may not reach the small gap within 4 hours or maximum iteration limits when datasets are larger than BM (249 samples).

Original: Recall that LD has no guarantee of reaching the global optimal solution because of the duality gap.
Revised: Recall that LD has no guarantee of reaching the global optimal solution because of the duality gap.

Original: All these results show that our method is extremely scalable for K-Medoids problems even under serial mode.
Revised: All these results show that our method is extremely scalable for K-Medoids problems even under serial mode.

Original: Notably, we followed the tuning guide in the CPLEX documentation to improve its performance.
Revised: Notably, we followed the tuning guide in the CPLEX documentation to improve its performance.

Original: For example, we initiated the root node with the Heuristic solution and specified the node file and memory parameters.
Revised: For example, we initiated the root node with the Heuristic solution and specified the node file and memory parameters.

Original: Moreover, we tried different MIP strategies, including start, pre-solve, and heuristic strategies.
Revised: Moreover, we tried different MIP strategies, including start, pre-solve, and heuristic strategies.

Original: In Table 1, we present the best results obtained by CPLEX.
Revised: In Table 1, we present the best results obtained by CPLEX.

Original: 6.2 Parallel Results
Revised: 6.2 Parallel Results

Original: For BB+LD, we executed parallel experiments utilizing MPI interfaces in the large and huge scale datasets with K = 3 .
Revised: For BB+LD, we executed parallel experiments utilizing MPI interfaces in the large and huge scale datasets with K = 3 .

Original: The results of large-scale datasets ranging from 10,000 to 100,000 samples are presented in Appendix B. Table 2 shows the parallel numerical results for huge datasets with 100, to 2,458,285 samples.
Revised: The results of large-scale datasets ranging from 10,000 to 100,000 samples are presented in Appendix B. Table 2 shows the parallel numerical results for huge datasets with 100, to 2,458,285 samples.

Original: From this table,we can see that even for the dataset with 1,046,910 samples, BB+LD can still converge to the global optimal solution ( ≤ 0.10 gaps) within one hour, which is 400 times larger than the size of problem that the state-of-art global solver CPLEX can address.
Revised: In this table, even for the dataset with 1,046,910 samples, BB+LD can still converge to the global optimal solution ( ≤ 0.10 gaps) within one hour, which is 400 times larger than the size of problem that the state-of-art global solver CPLEX can address.

Original: For USC1990 with 2,458,285 samples and 68 features, BB+LD can obtain a 6.33% gap withinhours.
Revised: For USC1990 with 2,458,285 samples and 68 features, BB+LD can obtain a 6.33% gap within 4 hours.

Original: According to Theorem 1, we can obtain a global ϵ -optimal solution for USC1990 given more calculation time
Revised: According to Theorem 1, we can obtain a global ϵ -optimal solution for USC1990 given more calculation time

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original:  Complexity Analysis 7.1 Time Complexity
Revised:  Complexity Analysis 7.1 Time Complexity

Original: Denoting the number of samples as S , attributes as A , and clusters as K
Revised: Denoting the number of samples as S , attributes as A [Table] and clusters as K

Original: [Table]
Revised: [Table]

Original: our algorithm branches on the medoid regions to guarantee convergence, which contains A × K variables and is independent of the number of samples S .
Revised: our algorithm branches on the medoid regions to guarantee convergence, which contains A × K variables and is independent of the number of samples S .

Original: However, the number of BB nodes to converge is hard to predict since the efficiency of bounding methods depends on datasets.
Revised: However, the number of BB nodes to converge is hard to predict since the efficiency of bounding methods depends on datasets.

Original: Also, the number of Lagrangian Dual (LD) iterations in each BB node is different because we set dynamic stopping criteria in case there is no update of LB in several continuous LD iterations.
Revised: Also, the number of Lagrangian Dual (LD) iterations in each BB node is different because we set dynamic stopping criteria in case there is no update of LB in several continuous LD iterations.

Original: The time complexity of one LD iteration is O ( S 2 ) when d s,j are precomputed.
Revised: The time complexity of one LD iteration is O ( S 2 ) when d s,j are precomputed.

Original: If d s,j are not precomputed, the time complexity is O ( AS 2 ) .
Revised: If d s,j are not precomputed, the time complexity is O ( AS 2 ) .

Original: Table 3 compares computing time and nodes on synthetic datasets with various dimensions and samples.
Revised: Table 3 compares computing time and nodes on synthetic datasets.

Original: In this table, the average run-time of one LD iteration remains almost the same as the dimension changes and increases almost quadratically as the number of samples increases.
Revised: In this table, the average runtime of one LD iteration remains almost the same as the dimension changes and increases almost quadratically as the number of samples increases.

Original: As for the BB nodes, although the number of branching variables is A × K , the actual number of explored BB nodes does not strictly follow the growth of dimensions.
Revised: As for the BB nodes, although the number of branching variables is A × K , the actual number of explored BB nodes does not strictly follow the growth of dimensions.

Original: These results are consistent with the complexity analysis mentioned above.
Revised: These results are consistent with the complexity analysis mentioned above.

Original: [Table]
Revised: [Table]

Original: 7.2 Space Complexity of Distance
Revised: 7.2 Space Complexity of Distance

Original: When distances d s,j are precomputed and stored
Revised: When distances d s,j are precomputed and stored

Original: [Table]
Revised: [Table]

Original: the space complexity of d s,j matrix is O ( S 2 ) .
Revised: the space complexity of d s [Table] j matrix is O ( S 2 ) .

Original: In our implementation
Revised: In our implementation

Original: [Table]
Revised: [Table]

Original: the datasets with no more than 100,000 samples were computed on one compute node with 40 cores and 202GB RAM.
Revised: the datasets with no more than 100,000 samples were computed on one compute node with 40 cores and 202GB RAM.

Original: For more than 100,000 samples, we executed the experiments on multiple compute nodes.
Revised: For more than 100,000 samples, we executed the experiments on multiple compute nodes.

Original: Each core precomputes and stores part of the d s,j matrix, as described in
Revised: Each core precomputes and stores part of the d s,j matrix, as described in

Original: Section 3.
Revised: Section 3.

Original: For example, we executed the one-million dataset RETAIL-II on 150 nodes with 6, cores.
Revised: For example, we executed the one-million dataset RETAIL-II on 150 nodes with 6, cores.

Original: Each core only needs to store 1.25GB of d s,j matrix.
Revised: Each core only needs to store 1.25GB of d s,j matrix.

Original: For more than 1,000,000 samples, we calculated d s,j on the fly, without precomputing and storing.
Revised: For more than 1,000,000 samples, we calculated d s,j on the fly, without precomputing and storing.

Original: In this case, the time complexity of one
Revised: In this case, the time complexity of one

Original: LD iteration increases from O ( S 2 ) to O ( AS 2 ) .
Revised: LD iteration increases from O ( S 2 ) to O ( AS 2 ) .

Original: Hence, we expect an acceptable slowdown when the dimension of datasets is small.
Revised: Hence, we expect an acceptable slowdown when the dimension of datasets is small.

Original: For example, in Table 2, the dataset USC1990 with two-million samples and 68 features was executed without precomputing d s,j , and the result is acceptable.
Revised: For example, in Table 2, the dataset USC1990 with two-million samples and 68 features was executed without precomputing d s,j , and the result is acceptable.

Original: Table 4 compares the computing time of precomputed distance and on-the-fly computing.
Revised: Table 4 compares the computing time of precomputed distance and on-the-fly computing.

Original: We can conclude that although calculating d s,j on the fly is slower than precomputing d s,j , the slowdown is acceptable.
Revised: We can conclude that although calculating d s,j on the fly is slower than precomputing d s,j , the slowdown is acceptable.

Original: [Table]
Revised: [Table]

Original:  Conclusion
Revised:  Conclusion

Original: We presented a scalable global optimization algorithm of the K-Medoids problem by applying a tailored reduced-space spatial branch and bound scheme.
Revised: We presented a scalable global optimization algorithm of the K-Medoids problem by applying a tailored reduced-space spatial branch and bound scheme.

Original: This algorithm includes a Lagrangian based lower bounding method and a basic lower bounding method.
Revised: This algorithm includes a Lagrangian based lower bounding method and a basic lower bounding method.

Original: Bound tightening techniques are also proposed to accelerate the solution process.
Revised: Bound tightening techniques are also proposed to accelerate the solution process.

Original: We demonstrate our algorithm’s scalability by extensive numerical experiments and prove the convergence by theoretical analysis.
Revised: We demonstrate our algorithm’s scalability by extensive numerical experiments and prove the convergence by theoretical analysis.

Original: Besides, one interesting finding is that our algorithm can reduce the objective values for some datasets while remaining the same as the heuristic method for most datasets.
Revised: Besides, one interesting finding is that our algorithm can reduce the objective values for some datasets while remaining the same as the heuristic method for most datasets.

Original: We also compared other clustering evaluation metrics in Appendix D, such as Normalized Mutual Information (NMI) and Adjusted Rand Index (ARI).
Revised: We also compared other clustering evaluation metrics in Appendix D, such as Normalized Mutual Information (NMI) and Adjusted Rand Index (ARI).

Original: A similar conclusion with objective values holds for these metrics.
Revised: A similar conclusion with objective values holds for these metrics.

Original: However, it should be noted that only with the global optimal solution obtained in our work can we give a fair comparison of the heuristic methods and, in turn, contribute to developing a more efficient heuristic method.
Revised: However, it should be noted that only with the global optimal solution obtained in our work can we give a fair comparison of the heuristic methods and, in turn, contribute to developing a more efficient heuristic method.

Original: Acknowledgement
Revised: Acknowledgments and Disclosure of Funding

Original: The authorsgratefully acknowledge  the computing resources provided by SciNet (www.scinethpc.ca) and Digital Research Alliance of Canada (www.alliancecan.ca).
Revised: The authors acknowledge funding from the discovery program of the Natural Science and Engineering Research Council of Canada under grant RGPIN-2019-05499 and the computing resources provided by SciNet (www.scinethpc.ca) and Digital Research Alliance of Canada (www.alliancecan.ca).

Original: Jiayang Ren acknowledges the financial support from the China Scholarship Council (Grant No. 202106320032).
Revised: Jiayang Ren acknowledges the financial support from the China Scholarship Council.

Original: Appendix A Convergence Analysis
Revised: Appendix A Convergence Analysis

Original: As stated in Algorithm 1, the branch and bound scheme is a rooted tree, where the search space of the level 0 root node is ◦ M .
Revised: As stated in Algorithm 1, the branch and bound scheme is a rooted tree, where the search space of the level 0 root node is ◦ M .

Original: We denote M l q as a sub-node at iteration l q and level q .
Revised: We denote M l q as a sub-node at iteration l q and level q .

Original: Its child node is denoted as M l q +1 satisfying M l q +1 ⊂ M l q .
Revised: Its child node is denoted as M l q +1 satisfying M l q +1 ⊂ M l q .

Original: A decreasing sequence from the root node ◦ M to the node M l q is denoted as { M l q } .
Revised: A decreasing sequence from the root node ◦ M to the node M l q is denoted as { M l q } .

Original: It is obvious that the sequences { α l } and { β l } are monotonically non-increasing and non-decreasing correspondingly.
Revised: It is obvious that the sequences { α l } and { β l } are monotonically non-increasing and non-decreasing correspondingly.

Original: In the following convergence analysis, we adapt the fundamental results from [29] to our work.
Revised: In the following convergence analysis, we adapt the fundamental results from [29] to our work.

Original: Different from the BB procedure in [19], which is infinite, our algorithm has a finite BB procedure.
Revised: Different from the BB procedure in [19], which is infinite, our algorithm has a finite BB procedure.

Original: Hence, we prove the convergence in a finite manner.
Revised: Hence, we prove the convergence in a finite manner.

Original: Definition 3.
Revised: Definition 3.

Original: (Definition IV.3 [29])
Revised: (Definition IV.3 [29])

Original: A bounding operation is called finitely consistent if, at every step, any unfathomed partition element can be further refined, and if any decreasing sequence { M l q } successively refined partition elements is finite.
Revised: A bounding operation is called finitely consistent if, at every step, any unfathomed partition element can be further refined, and if any decreasing sequence { M l q } successively refined partition elements is finite.

Original: Lemma 4.
Revised: Lemma 4.

Original: The bounding operation in Algorithm 1 is finitely consistent.
Revised: The bounding operation in Algorithm 1 is finitely consistent.

Original: Proof.
Revised: Proof.

Original: We first prove any unfathomed partition element M l q can be further refined.
Revised: We first prove any unfathomed partition element M l q can be further refined.

Original: Any unfathomed
Revised: Any unfathomed

Original: M l q satisfies ∃| M l q ∩ X | > 1 , k ∈ K and α l > β ( M l q ) + ϵ, ϵ > 0 .
Revised: M l q satisfies ∃| M l q ∩ X | > 1 , k ∈ K and α l > β ( M l q ) + ϵ, ϵ > 0 .

Original: It is obvious that there exists at least one partition to be further refined.
Revised: It is obvious that there exists at least one partition to be further refined.

Original: We then prove any decreasing sequence { M l q } successively refined partition elements is finite.
Revised: We then prove any decreasing sequence { M l q } successively refined partition elements is finite.

Original: Assuming by contradiction that there exists a sequence { M l q } that is infinite.
Revised: Assuming by contradiction that there exists a sequence { M l q } that is infinite.

Original: In our algorithm, since we branch on the first-stage variable µ k corresponding to the diameter of M , this subdivision is exhaustive, we have lim q →∞ δ
Revised: In our algorithm, since we branch on the first-stage variable µ k corresponding to the diameter of M , this subdivision is exhaustive, we have lim q →∞ δ

Original: ( M l q )
Revised: ( M l q )

Original: = 

Revised: =

Original: and { M l q } converge to one point ¯ µ .
Revised: and { M l q } converge to one point ¯ µ .

Original: If ¯ µ ∈ X , there is a ball around ¯ µ , denoted as B r (¯ µ ) =
Revised: If ¯ µ ∈ X , there is a ball around ¯ µ , denoted as B r (¯ µ ) =

Original: { µ | || µ − ¯ µ || ≤ r } , satisfying | B r (¯ µ ) ∩ X | = 1 .
Revised: { µ | || µ − ¯ µ || ≤ r } , satisfying | B r (¯ µ ) ∩ X | = 1 .

Original: There exists a q 0 that M l q ⊂ B r (¯ µ ) , ∀ q ≤ q 0 .
Revised: There exists a q 0 that M l q ⊂ B r (¯ µ ) , ∀ q ≤ q 0 .

Original: At this q 0 iteration, M l q 0 will not be branched anymore.
Revised: At this q 0 iteration, M l q 0 will not be branched anymore.

Original: Because
Revised: Because

Original: X is finite, we have the sequence { M l q } is finite in this case.
Revised: X is finite, we have the sequence { M l q } is finite in this case.

Original: If ¯ µ ̸⊂ X , there is a ball around ¯ µ , denoted as B r (¯ µ ) =
Revised: If ¯ µ ̸⊂ X , there is a ball around ¯ µ , denoted as B r (¯ µ ) =

Original: { µ | || µ − ¯ µ || ≤ r } , satisfying | B r (¯ µ ) ∩ X | = 0 .
Revised: { µ | || µ − ¯ µ || ≤ r } , satisfying | B r (¯ µ ) ∩ X | = 0 .

Original: There exists a q 0 that
Revised: There exists a q 0 that

Original: M l q ⊂ B r (¯ µ ) , ∀ q ≤ q 0 .
Revised: M l q ⊂ B r (¯ µ ) , ∀ q ≤ q 0 .

Original: At this q 0 iteration, M l q 0 will be deleted.
Revised: At this q 0 iteration, M l q 0 will be deleted.

Original: Consequently, in this case, the sequence { M l q } is finite.
Revised: Consequently, in this case, the sequence { M l q } is finite.

Original: Hence, it is impossible to exists a sequence { M l q } that is infinite.
Revised: Hence, it is impossible to exists a sequence { M l q } that is infinite.

Original: Theorem 5.
Revised: Theorem 5.

Original: (Theorem IV.1 [29])
Revised: (Theorem IV.1 [29])

Original: In a BB procedure, suppose that the bounding operation is finitely consistent.
Revised: In a BB procedure, suppose that the bounding operation is finitely consistent.

Original: Then the procedure terminates after finitely many steps.
Revised: Then the procedure terminates after finitely many steps.

Original: Lemma 6.
Revised: Lemma 6.

Original: Algorithm 1 terminates after finitely many steps
Revised: Algorithm 1 terminates after finitely many steps

Original: Proof.
Revised: Proof.

Original: From Lemma 4, we have the bounding operation in Algorithm 1 is finitely consistent.
Revised: From Lemma 4, we have the bounding operation in Algorithm 1 is finitely consistent.

Original: From
Revised: From

Original: Theorem 5, Algorithm 1 terminates after finitely many steps
Revised: Theorem 5, Algorithm 1 terminates after finitely many steps

Original: Finally, we prove that the BB scheme is convergent, as shown in Theorem 1:
Revised: Finally, we prove that the BB scheme is convergent, as shown in Theorem 1:

Original: Theorem 1.
Revised: Theorem 1.

Original: Algorithm 1 is convergent to the global optimal solution after a finite step L , with β L =
Revised: Algorithm 1 is convergent to the global optimal solution after a finite step L , with β L =

Original: z = α L , by only branching on the space of µ .
Revised: z = α L , by only branching on the space of µ .

Original: Proof.
Revised: Proof.

Original: From Lemma 6, we have Algorithm 1 terminates after finite steps.
Revised: From Lemma 6, we have Algorithm 1 terminates after finite steps.

Original: The algorithm terminates with two situations.
Revised: The algorithm terminates with two situations.

Original: The first situations is | β l − α l | ≤ ϵ .
Revised: The first situations is | β l − α l | ≤ ϵ .

Original: When ϵ = 0 , we have β l = z = α l .
Revised: When ϵ = 0 , we have β l = z = α l .

Original: The second situations is M = ∅ .
Revised: The second situations is M = ∅ .

Original: A node M is deleted from M and not further partitioned either because β ( M ) >
Revised: A node M is deleted from M and not further partitioned either because β ( M ) >

Original: α l or | M k ∩ X | = 1 , ∀ k ∈ K .
Revised: α l or | M k ∩ X | = 1 , ∀ k ∈ K .

Original: The first case obviously does not contain the global optimal solution µ ∗ .
Revised: The first case obviously does not contain the global optimal solution µ ∗ .

Original: Therefore, the node M ′ containing µ ∗ , is not further partitioned because | M ′ k
Revised: Therefore, the node M ′ containing µ ∗ , is not further partitioned because | M ′ k

Original: ∩ X | = 1 , ∀ k ∈ K .
Revised: ∩ X | = 1 , ∀ k ∈ K .

Original: After bound tightening according to the “medoids on samples” constraint, the tightened node M ′
Revised: After bound tightening according to the “medoids on samples” constraint, the tightened node M ′

Original: = { µ ∗ } .
Revised: = { µ ∗ } .

Original: Obviously for this node, we have β l = β ( M ′ )
Revised: Obviously for this node, we have β l = β ( M ′ )

Original: =
Revised: =

Original: z = α ( M ′ )
Revised: z = α ( M ′ )

Original: =
Revised: =

Original: α l .
Revised: α l .

Original: Consequently, we have proved Theorem 1.
Revised: Consequently, we have proved Theorem 1.

Original: B Parallel results of large scale datasets
Revised: B Parallel results of large scale datasets

Original: Table 5 shows the parallel numerical results for large scale datasets with 10,000 to 2,458,285 samples.
Revised: Table 5 shows the parallel numerical results for large scale datasets with 10,000 to 2,458,285 samples.

Original: For the large-scale datasets ranging from 10,000 to 100,000 samples, we obtain an average of 13.80x speedup in time with 40 cores.
Revised: For the large-scale datasets ranging from 10,000 to 100,000 samples, we obtain an average of 13.80x speedup in time with 40 cores.

Original: Here, speedup ratio is defined as ¯ T 1 ¯ T c , in which ¯ T c is the overall calculation time with c cores.
Revised: Here, speedup ratio is defined as ¯ T 1 ¯ T c , in which ¯ T c is the overall calculation time with c cores.

Original: The average efficiency is low since the time ratio of the parallel part is relatively low in the small datasets.
Revised: The average efficiency is low since the time ratio of the parallel part is relatively low in the small datasets.

Original: Consequently, we can expect a higher speedup in the big datasets.
Revised: Consequently, we can expect a higher speedup in the big datasets.

Original: For example, we obtain a 25.89x Speedup for URBANGB_10 with 100,000 samples compared to 7.57x for HTRU2 with 17,898 samples.
Revised: For example, we obtain a 25.89x Speedup for URBANGB_10 with 100,000 samples compared to 7.57x for HTRU2 with 17,898 samples.

Original: [Table]
Revised: [Table]

Original: C Additional results of K-Medoids problems with K = 5 and K =
Revised: C Additional results of K-Medoids problems with K = 5 and K =

Original: In this section, we perform several additional experiments with more clusters on the datasets ranging from 100 to 100,000 samples.
Revised: In this section, we perform several additional experiments with more clusters on the datasets ranging from 100 to 100,000 samples.

Original: All these experiments use the same setup in Section 6.
Revised: All these experiments use the same setup in Section 6.

Original: Table 6 andare the results of the BB+LD method with cluster number K = 5 and K = 10 respectively.
Revised: Table 6 andare the results of the BB+LD method with cluster number K = 5 and K = 10 respectively.

Original: When the cluster number is bigger, our BB+LD are more likely to obtain an upper bound smaller than the Heuristic UB.
Revised: When the cluster number is bigger, our BB+LD are more likely to obtain an upper bound smaller than the Heuristic UB.

Original: However, the difference in UB between BB+LD and Heuristic becomes smaller.
Revised: However, the difference in UB between BB+LD and Heuristic becomes smaller.

Original: Moreover, the search space of medoids increases as the cluster number, making it harder to obtain the global optimum.
Revised: Moreover, the search space of medoids increases as the cluster number, making it harder to obtain the global optimum.

Original: Nevertheless, our BB+LD method can still obtain a relative gap smaller than 0.1% within 4 hours for most datasets when K = 5 and a reasonable relative gap within 4 hours when
Revised: Nevertheless, our BB+LD method can still obtain a relative gap smaller than 0.1% within 4 hours for most datasets when K = 5 and a reasonable relative gap within 4 hours when

Original: K =
Revised: K =

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table]
Revised: [Table]

Original: D Comparison of heuristic methods for K-Medoids problems
Revised: D Comparison of heuristic methods for K-Medoids problems

Original: To illustrate the effectiveness of our upper bound method (Heuristic UB)
Revised: To illustrate the effectiveness of our upper bound method (Heuristic UB)

Original: [Table]
Revised: [Table]

Original: we compares its performance with several popular heuristic methods in the literature, including Kmeans, Kmeans++, and PAM.
Revised: we compares its performance with several popular heuristic methods in the literature, including Kmeans, Kmeans++, and PAM.

Original: Here, we run all the methods several times with random seeds and select the best result.
Revised: Here, we run all the methods several times with random seeds and select the best result.

Original: The centers of K-Means and K-Means++ are projected to the nearest samples to fulfill the "Medoids on Samples" constraint in the KMedoids problem.
Revised: The centers of K-Means and K-Means++ are projected to the nearest samples to fulfill the "Medoids on Samples" constraint in the KMedoids problem.

Original: This table shows that the Heuristic UB can always obtain the same or better objective value than Kmeans, Kmeans++, and PAM.
Revised: This table shows that the Heuristic UB can always obtain the same or better objective value than Kmeans, Kmeans++, and PAM.

Original: [Table]
Revised: [Table]

Original: We also compared other clustering evaluation metrics, such as Normalized Mutual Information (NMI) and Adjusted Rand Index (ARI).
Revised: We also compared other clustering evaluation metrics, such as Normalized Mutual Information (NMI) and Adjusted Rand Index (ARI).

Original: Since these clustering evaluation metrics need to compare the predicted cluster with ground truth, we can only compare the datasets with ground truth, such as
Revised: Since these clustering evaluation metrics need to compare the predicted cluster with ground truth, we can only compare the datasets with ground truth, such as

Original: IRIS, HCV, HF, HRTU2 and UK.
Revised: IRIS, HCV, HF, HRTU2 and UK.

Original: Comparing Table 8 and Table 9, one interesting finding is that our algorithm can reduce the objective values for some datasets while remaining the same as the heuristic method for most datasets.
Revised: Comparing Table 8 and Table 9, one interesting finding is that our algorithm can reduce the objective values for some datasets while remaining the same as the heuristic method for most datasets.

Original: A similar conclusion with objective values also holds for ARI and NMI.
Revised: A similar conclusion with objective values also holds for ARI and NMI.

Original: However, it should be noted that only by comparing the heuristic solution with the global optimal solution as we did in the table
Revised: However, it should be noted that only by comparing the heuristic solution with the global optimal solution as we did in the table

Original: [Table]
Revised: [Table]

Original: we can confidently claim that the heuristic method can do a fairly nice job in finding near-optimal solutions
Revised: we can confidently claim that the heuristic method can do a fairly nice job in finding near-optimal solutions

Original: [Table] [Table]
Revised: [Table] [Table]