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Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning”
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Short Communication
Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning” Janiele Custodio a, Miguel Lejeune b, Antonio Zavaleta a,∗ a
Department of Engineering Management and Systems Engineering, The George Washington University, 800 22nd St NW, Washington, DC 20052, United States b Decision Sciences Department, The George Washington University, 2201 G Street NW Suite 406, Washington, DC 20052, United States
a r t i c l e
i n f o
Article history: Received 25 May 2018 Accepted 28 November 2018 Available online xxx Keywords: OR in health services Chance-constrained programming Monotonic transformation Reformulation
a b s t r a c t A recent article by Zaghian et al. (2018) proposed reformulations, proposed reformulations of stochastic chance-constrained programming models for radiation therapy treatment planning. This note questions the validity of the proposed reformulations and shows that they are not equivalent to the original formulations. Two numerical examples illustrate that the approach proposed by Zaghian et al. (2018) provides approximation problems and not reformulations.
1. Introduction This note clarifies the results presented in a recent study by Zaghian, Lim, and Khabazian (2018), who propose a reformulation approach a reformulation approach for chance-constrained programming (CCP) models used for radiation therapy treatment planning. It is generally understood that the reformulation of an optimization problem is such that the optimal objective value and optimal solutions (if they exist) of one problem can be easily obtained from the optimal objective value and optimal solutions of the other problem, and vice versa (Calafiore and El Ghaoui, 2014, p.257). The two problems are said to be equivalent. In such case, either the optimal solution sets for both problems are identical or there exists a bijection from one solution set to the other. In light of this, we question the validity of presenting (see Sections 3.2.1 and 3.3.1 in Zaghian et al., 2018): •
•
Model (20) (Zaghian et al., 2018, p.739) as a reformulation of problem (19) (Zaghian et al., 2018, p.739) when normal distributions are assumed. Model (31) (Zaghian et al., 2018, p.741) as a reformulation of problem (19) (Zaghian et al., 2018, p.739) when Beta distributions are assumed.
In the following sections, we show that problems (20) and (31) are not equivalent reformulations of the chance-constrained prob∗
Corresponding author. E-mail addresses: [email protected] (J. Custodio), [email protected] (M. Lejeune), [email protected], [email protected] (A. Zavaleta).
Published by Elsevier B.V.
lem (19) in Zaghian et al. (2018). There is no guarantee that the optimal solutions of (20) and (31) can be directly mapped to the optimal solution of (19). 2. Chance-constrained problems and reformulated models The optimization model (19) proposed in Zaghian et al. (2018) is the equivalent deterministic reformulation of a CCP problem. In this type of formulations, it is commonplace to have constraints involving the quantile function of a probability distribution (Prékopa, 1995). The presence of nonlinear (often not nonconvex) quantile functions can make the solution of the corresponding optimization problem challenging. This motivates Zaghian et al. (2018) to derive a formulation that is more easily amenable to a numerical solution. To accomplish this, they replace the objective function
max zO,G =
3
λi (1 − αi )
(1)
i=1
in the original problem (19) by
max zF,G =
3
λi −1 (1 − αi )
(2)
i=1
in the so-called reformulated problem (20) and keep the constraint set unchanged. The notation −1 (· ) denotes the quantile or inverse of the standard normal distribution function, λ ∈ R3+ is a vector of fixed parameters, and α ∈ [0, 0.5]3 is a vector of decision variables defining confidence levels.
https://doi.org/10.1016/j.ejor.2018.11.071 0377-2217/Published by Elsevier B.V.
Please cite this article as: J. Custodio, M. Lejeune and A. Zavaleta, Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning”, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2018.11. 071
ARTICLE IN PRESS
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[m5G;December 14, 2018;16:8]
J. Custodio, M. Lejeune and A. Zavaleta / European Journal of Operational Research xxx (xxxx) xxx
Similarly, by combining the objective function provided in (19) with the constraint set (23)–(25) in Zaghian et al. (2018), a new optimization model (31) is obtained when random variables are assumed to follow a Beta distribution. This model is similar to (19) save for the Beta probability distributions. The objective function for this problem will be referred to as zO, B . The proposed alternative model (31) has the following objective function
max zF,B =
3
λi κ(αi )
(3)
i=1
−1 υ 2 beta ( 1 − 2α )
with κ(α ) = for α ∈ [0; where is the inverse distribution of a Beta ( 1 / 2 ; n/ 2 + 1 ) random variable, and √ υ = n + 3. In general, the application of a monotonic transformation to the objective function seeks to maintain or create convexity in an optimization problem while maintaining the feasibility set unchanged. Next, we describe the process of applying a monotonic transformation to the objective function (Calafiore & El Ghaoui, 2014). Consider a convex optimization problem of form
p∗ = min s.to
0.5]3 ,
−1 beta (· )
f 0 (x ) f i ( x ) ≤ 0, i = 1, . . . , m
(4)
Let ϕ : R → R be a continuous and strictly increasing function used to derive the transformed optimization problem:
g∗ = min ϕ ( f0 (x )) s.to fi (x ) ≤ 0
i = 1, . . . , m
(5)
Problems (4) and (5) have the same feasible set and have identical optimal solutions x∗ . The optimal solution of (4) can be retrieved from that of (5) (and vice-versa) as follows:
ϕ (g ) = f 0 (x ) = p −1
∗
∗
∗
(6)
Zaghian et al. (2018) justify using the transformed objective function (2) by the fact that the inverse normal distribution −1 (z ) is monotonically increasing in z. However, while −1 (z ) is isotonic in z, the proposed objective function does not result from a transformation as defined in (5). Instead, the reformulation approach presented in Zaghian et al. (2018) is obtained by separately applying −1 (· ) to each element of the vector α in (1). The resulting reformulation (2) is not isotonic1 . To see this, it suffices to show that there exists at least two vectors, say α (1) and α (2) , such that
zO,G (α (1) ) ≤ zO,G (α (2) )
and zF,G (α (1) ) > zF,G (α (2) ) .
Table 1 Numerical results for problem with normal distribution: λi = 1 ∀i, μ = 0, σ = 1. Value
zO, G (α )
Rank zO, G (α )
zF, G (α )
Rank zF, G (α )
α (1) = [0.3125 0.0176 0.1259] α (2) = [0.1615 0.103 0.1185]
2.5440 2.6170
2 1
3.7408 3.4355
1 2
Table 2 Numerical results for problem with beta distribution: λi = 1 ∀i, n = 10, α = 1/2, β = n/2 + 1. Value
zO, G (α )
Rank zO, G (α )
zF, B (α )
Rank zF, B (α )
α (1 )
2.3652
2 1
6.5142 6.1848
1 2
= [0.0073 0.4886 0.1389]
α (2) = [0.0486 0.0568 0.1839] 2.7107
Table 1 presents a numerical example for the transformation proposed by Zaghian et al. (2018) when the dose contribution vector d˜i is assumed to be normally distributed. Note that, while
zO, G (α (1) ) > zO, G (α (2) ), zF, G (α (1) ) < zF, G (α (2) ), which is the contradiction described in (7). This shows that (1) is not a monotonic mapping of (2). Therefore, the optimal solution(s) of problem (19) and (20) in Zaghian et al. (2018) are not guaranteed to be the same. Table 2 displays a numerical example for the transformation (31) in Zaghian et al. (2018), which assumes a Beta probability distribution. We reached the same results as above: zO, B (α (1) ) > zO, B (α (2) ), but zF, B (α (1) ) < zF, B (α (2) ), which indicates that (3) is not a monotonic transformation of the objective function (1) either. 4. Conclusion Zaghian et al. (2018) introduce CCP formulations for radiotherapy treatment planning. The authors propose alternative formulations ((20) and (31)) for problem (19) by using the monotonicity property of the normal and Beta inverse distribution functions. In this note, we rectify these statements and show that two alternative formulations proposed by Zaghian et al. (2018) are not equivalent to the original problem. It is difficult to evaluate by how much the optimal solution and value of the alternative formulations (20) and (31) differ form those of the original formulation (19). Acknowledgments
(7)
The existence of such two vectors implies that the transformed objective function (2) is not a monotonic mapping of the true objective function (1), which demonstrates that the formulations (19) and (20) in Zaghian et al. (2018) are not equivalent. The reasoning for Eq. (3) is similar, except for the use of Beta distributions instead of normal ones. We provide illustrative numerical examples for both cases in the next section. 3. Numerical examples This section presents two numerical examples showing that the proposed models (20) and (31) in Zaghian et al. (2018) are not equivalent to (19).
The authors would like to thank the editorial and review team for their thoughtful comments and recommendations. Lejeune acknowledges the support of the Office of Naval Research [Grant N0 0 0141712420].
References Calafiore, G., & El Ghaoui, L. (2014). Optimization models. Cambridge University Press. Prékopa, A. (1995). Stochastic programming. Springer Science. Zaghian, M., Lim, G., & Khabazian, A. (2018). A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning. European Journal of Operational Research, 266(2), 736–745.
1 The proposed transformation would be a proper reformulation only if −1 (· ) was a linear function with intercept equal to zero.
Please cite this article as: J. Custodio, M. Lejeune and A. Zavaleta, Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning”, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2018.11. 071
JID: EOR
[m5G;December 14, 2018;16:8]
European Journal of Operational Research xxx (xxxx) xxx
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Short Communication
Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning” Janiele Custodio a, Miguel Lejeune b, Antonio Zavaleta a,∗ a
Department of Engineering Management and Systems Engineering, The George Washington University, 800 22nd St NW, Washington, DC 20052, United States b Decision Sciences Department, The George Washington University, 2201 G Street NW Suite 406, Washington, DC 20052, United States
a r t i c l e
i n f o
Article history: Received 25 May 2018 Accepted 28 November 2018 Available online xxx Keywords: OR in health services Chance-constrained programming Monotonic transformation Reformulation
a b s t r a c t A recent article by Zaghian et al. (2018) proposed reformulations, proposed reformulations of stochastic chance-constrained programming models for radiation therapy treatment planning. This note questions the validity of the proposed reformulations and shows that they are not equivalent to the original formulations. Two numerical examples illustrate that the approach proposed by Zaghian et al. (2018) provides approximation problems and not reformulations.
1. Introduction This note clarifies the results presented in a recent study by Zaghian, Lim, and Khabazian (2018), who propose a reformulation approach a reformulation approach for chance-constrained programming (CCP) models used for radiation therapy treatment planning. It is generally understood that the reformulation of an optimization problem is such that the optimal objective value and optimal solutions (if they exist) of one problem can be easily obtained from the optimal objective value and optimal solutions of the other problem, and vice versa (Calafiore and El Ghaoui, 2014, p.257). The two problems are said to be equivalent. In such case, either the optimal solution sets for both problems are identical or there exists a bijection from one solution set to the other. In light of this, we question the validity of presenting (see Sections 3.2.1 and 3.3.1 in Zaghian et al., 2018): •
•
Model (20) (Zaghian et al., 2018, p.739) as a reformulation of problem (19) (Zaghian et al., 2018, p.739) when normal distributions are assumed. Model (31) (Zaghian et al., 2018, p.741) as a reformulation of problem (19) (Zaghian et al., 2018, p.739) when Beta distributions are assumed.
In the following sections, we show that problems (20) and (31) are not equivalent reformulations of the chance-constrained prob∗
Corresponding author. E-mail addresses: [email protected] (J. Custodio), [email protected] (M. Lejeune), [email protected], [email protected] (A. Zavaleta).
Published by Elsevier B.V.
lem (19) in Zaghian et al. (2018). There is no guarantee that the optimal solutions of (20) and (31) can be directly mapped to the optimal solution of (19). 2. Chance-constrained problems and reformulated models The optimization model (19) proposed in Zaghian et al. (2018) is the equivalent deterministic reformulation of a CCP problem. In this type of formulations, it is commonplace to have constraints involving the quantile function of a probability distribution (Prékopa, 1995). The presence of nonlinear (often not nonconvex) quantile functions can make the solution of the corresponding optimization problem challenging. This motivates Zaghian et al. (2018) to derive a formulation that is more easily amenable to a numerical solution. To accomplish this, they replace the objective function
max zO,G =
3
λi (1 − αi )
(1)
i=1
in the original problem (19) by
max zF,G =
3
λi −1 (1 − αi )
(2)
i=1
in the so-called reformulated problem (20) and keep the constraint set unchanged. The notation −1 (· ) denotes the quantile or inverse of the standard normal distribution function, λ ∈ R3+ is a vector of fixed parameters, and α ∈ [0, 0.5]3 is a vector of decision variables defining confidence levels.
https://doi.org/10.1016/j.ejor.2018.11.071 0377-2217/Published by Elsevier B.V.
Please cite this article as: J. Custodio, M. Lejeune and A. Zavaleta, Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning”, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2018.11. 071
ARTICLE IN PRESS
JID: EOR 2
[m5G;December 14, 2018;16:8]
J. Custodio, M. Lejeune and A. Zavaleta / European Journal of Operational Research xxx (xxxx) xxx
Similarly, by combining the objective function provided in (19) with the constraint set (23)–(25) in Zaghian et al. (2018), a new optimization model (31) is obtained when random variables are assumed to follow a Beta distribution. This model is similar to (19) save for the Beta probability distributions. The objective function for this problem will be referred to as zO, B . The proposed alternative model (31) has the following objective function
max zF,B =
3
λi κ(αi )
(3)
i=1
−1 υ 2 beta ( 1 − 2α )
with κ(α ) = for α ∈ [0; where is the inverse distribution of a Beta ( 1 / 2 ; n/ 2 + 1 ) random variable, and √ υ = n + 3. In general, the application of a monotonic transformation to the objective function seeks to maintain or create convexity in an optimization problem while maintaining the feasibility set unchanged. Next, we describe the process of applying a monotonic transformation to the objective function (Calafiore & El Ghaoui, 2014). Consider a convex optimization problem of form
p∗ = min s.to
0.5]3 ,
−1 beta (· )
f 0 (x ) f i ( x ) ≤ 0, i = 1, . . . , m
(4)
Let ϕ : R → R be a continuous and strictly increasing function used to derive the transformed optimization problem:
g∗ = min ϕ ( f0 (x )) s.to fi (x ) ≤ 0
i = 1, . . . , m
(5)
Problems (4) and (5) have the same feasible set and have identical optimal solutions x∗ . The optimal solution of (4) can be retrieved from that of (5) (and vice-versa) as follows:
ϕ (g ) = f 0 (x ) = p −1
∗
∗
∗
(6)
Zaghian et al. (2018) justify using the transformed objective function (2) by the fact that the inverse normal distribution −1 (z ) is monotonically increasing in z. However, while −1 (z ) is isotonic in z, the proposed objective function does not result from a transformation as defined in (5). Instead, the reformulation approach presented in Zaghian et al. (2018) is obtained by separately applying −1 (· ) to each element of the vector α in (1). The resulting reformulation (2) is not isotonic1 . To see this, it suffices to show that there exists at least two vectors, say α (1) and α (2) , such that
zO,G (α (1) ) ≤ zO,G (α (2) )
and zF,G (α (1) ) > zF,G (α (2) ) .
Table 1 Numerical results for problem with normal distribution: λi = 1 ∀i, μ = 0, σ = 1. Value
zO, G (α )
Rank zO, G (α )
zF, G (α )
Rank zF, G (α )
α (1) = [0.3125 0.0176 0.1259] α (2) = [0.1615 0.103 0.1185]
2.5440 2.6170
2 1
3.7408 3.4355
1 2
Table 2 Numerical results for problem with beta distribution: λi = 1 ∀i, n = 10, α = 1/2, β = n/2 + 1. Value
zO, G (α )
Rank zO, G (α )
zF, B (α )
Rank zF, B (α )
α (1 )
2.3652
2 1
6.5142 6.1848
1 2
= [0.0073 0.4886 0.1389]
α (2) = [0.0486 0.0568 0.1839] 2.7107
Table 1 presents a numerical example for the transformation proposed by Zaghian et al. (2018) when the dose contribution vector d˜i is assumed to be normally distributed. Note that, while
zO, G (α (1) ) > zO, G (α (2) ), zF, G (α (1) ) < zF, G (α (2) ), which is the contradiction described in (7). This shows that (1) is not a monotonic mapping of (2). Therefore, the optimal solution(s) of problem (19) and (20) in Zaghian et al. (2018) are not guaranteed to be the same. Table 2 displays a numerical example for the transformation (31) in Zaghian et al. (2018), which assumes a Beta probability distribution. We reached the same results as above: zO, B (α (1) ) > zO, B (α (2) ), but zF, B (α (1) ) < zF, B (α (2) ), which indicates that (3) is not a monotonic transformation of the objective function (1) either. 4. Conclusion Zaghian et al. (2018) introduce CCP formulations for radiotherapy treatment planning. The authors propose alternative formulations ((20) and (31)) for problem (19) by using the monotonicity property of the normal and Beta inverse distribution functions. In this note, we rectify these statements and show that two alternative formulations proposed by Zaghian et al. (2018) are not equivalent to the original problem. It is difficult to evaluate by how much the optimal solution and value of the alternative formulations (20) and (31) differ form those of the original formulation (19). Acknowledgments
(7)
The existence of such two vectors implies that the transformed objective function (2) is not a monotonic mapping of the true objective function (1), which demonstrates that the formulations (19) and (20) in Zaghian et al. (2018) are not equivalent. The reasoning for Eq. (3) is similar, except for the use of Beta distributions instead of normal ones. We provide illustrative numerical examples for both cases in the next section. 3. Numerical examples This section presents two numerical examples showing that the proposed models (20) and (31) in Zaghian et al. (2018) are not equivalent to (19).
The authors would like to thank the editorial and review team for their thoughtful comments and recommendations. Lejeune acknowledges the support of the Office of Naval Research [Grant N0 0 0141712420].
References Calafiore, G., & El Ghaoui, L. (2014). Optimization models. Cambridge University Press. Prékopa, A. (1995). Stochastic programming. Springer Science. Zaghian, M., Lim, G., & Khabazian, A. (2018). A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning. European Journal of Operational Research, 266(2), 736–745.
1 The proposed transformation would be a proper reformulation only if −1 (· ) was a linear function with intercept equal to zero.
Please cite this article as: J. Custodio, M. Lejeune and A. Zavaleta, Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning”, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2018.11. 071