Fundamentals of convex optimization for compositional data

SORT 47 (2) July-December 2023, 323-344 DOI: 10.57645/20.8080.02.11 Fundamentals of convex optimization for compositional data Jordi Saperas Riera1, Josep Antoni Mart´ ın Fern´ andez3 and Gl` oria Mateu Figueras2 Abstract Many of the most popular statistical techniques incorporate optimisation problems in their inner workings. A convex optimisation problem is defned as the problem of minimising a convex function over a convex set. When traditional methods are applied to compositional data, misleading and incoherent results could be obtained. In this paper, we fll a gap in the specialised literature by introducing and rigorously defning novel concepts of convex optimisation for compositional data according to the Aitchison geometry. Convex sets and convex functions on the simplex are defned and illustrated. MSC: 62F40, 62H15, 62H99, 62J10, 62J15, 62P25. Keywords: Compositional data, logratio, simplex, proportion, function, convexity, optimisation. 1. Introduction Convex optimisation plays an important role in a wide range of scientifc felds where statistical methods are applied. Thus, it has become particularly relevant in the design of experiments (Coetzer and Haines, 2017), variable selection (Susin et al., 2020), robust statistics (Boogaart et al., 2021), cluster analysis (Wang et al., 2020), and principal components analysis (Campbell and Wong, 2022). In a broad sense, a convex optimisation problem in RD is defned as (Boyd and Vandenberghe, 2004) the problem of minimising a convex function (objective function) over a convex set (feasible region). The feasible region is the set of all possible values of variables that verify the constraints of the problem. Commonly, it is assumed 1,2,3 Department of Computer Science, Applied Mathematics and Statistics, University of Girona, Campus Montilivi, Edifci P-4, 17003 Girona, Spain. 1 [email protected] 2 [email protected] 3 [email protected] Received: November 2022 Accepted: June 2023 324 Fundamentals of convex optimization for compositional data that the convexity of the objective function and the feasible region corresponds to the ordinary convexity concept in real space, RD . However, in statistics, one has to take into account the particular geometry of the sample space of the variables. Compositional data (CoDa) (Aitchison, 1986) convey relative information because the variables describe relative contributions to a given total. These variables are called parts of a whole and are, usually, expressed in proportions, percentages or ppm. Historically (Aitchison, 1986), the sample space of CoDa is designed as the D-part unit simplex SD = {x ∈ RD : xj > 0;∑xj = 1; j = 1,...,D}. The formal geometric framework for the analysis of CoDa was frst introduced in Pawlowsky-Glahn and Egozcue (2001) and Billheimer, Guttorp and Fagan (2001). It was called the Aitchison geometry, and it was later formally established in Barcelo-Vidal and Mart ´ ´ ın-Fern´ andez (2016). Such geometry allows compositions to be expressed as coordinates on an orthonormal basis, formed by logratios and called olr-coordinates (Egozcue et al., 2003; Mart´ ın-Fern´ andez, 2019). In short, the analysis of CoDa involves the use of standard techniques in olrcoordinates; what has been known as the principle of working on coordinates (MateuFigueras, Pawlowsky-Glahn and Egozcue, 2011). Applications of the log-ratio methodology are increasingly found across a wide range of scientifc felds such as geosciences (Mart´ ın-Fern´ andez et al., 2018), chemistry and physics (Halim et al., 2021), and health (Bates and Tibshirani, 2019; Dumuid et al., 2020), among others. CoDa methods have become particularly relevant for the analysis of time-use data, especially in relation to public health studies (Chastin et al., 2015; Dumuid et al., 2021; Kitano et al., 2020; Fairclough et al., 2018; Gupta et al., 2020). In time-use data, the parts are the time spent on different activities such as sleep, work, and a range of physical activities. Some optimisation problems of practical relevance can be formulated in this context. For example, one can be interested in deciding what composition of daily activities agreeing with some medical recommendations (constraints) optimises a particular health biomarker (objective function). Alternatively, given a timeuse composition of a patient not following some medical recommendation (feasible region), the question of interest is: what modifcation to the daily activity composition would enable a swift transition (objective function) into a healthy behaviour region? The aim of this work is to adapt the concepts related to convex optimisation for problems involving CoDa and considering the Aitchison geometry. These defnitions are essential to correctly identify and classify convex optimisation problems on the simplex. The paper is organised as follows. Section 2 summarises the basic concepts of CoDa. In Section 3, the concept of convex set on the simplex with the Aitchison geometry is defned. The basic sets on the simplex are analysed and their compositional convexity is studied. Section 4 develops the concept of compositional convex function and introduces the conditions to classify a problem as a compositional convex optimisation problem. These are illustrated in Section 5 through a case study based on time-use data. Finally, Section 6 concludes with some important remarks. The analyses discussed in this article were carried out in R (R-Core-Team, 2022) and using the package compositions (van den Boogaart and Tolosana-Delgado, 2008). Jordi Saperas Riera, Josep Antoni Mart´ ın Fernandez and Gl` 325 ´ oria Mateu Figueras 2. Basic elements of Aitchison geometry When analysing CoDa one assumes the property of scale invariance. That is, it is assumed that each D-part composition w ∈ RD is a member of an equivalence class (Bar- + cel´ o-Vidal and Mart´ ın-Fern´ andez, 2016). In other words, the information contained in w is the same as in any other composition k · C[w] for any real scalar k > 0, where C[w] is the closure operation defned by C[w] = [w1/∑wj,w2/∑wj,...,wD/∑wj] = x ∈ SD . The perturbation operation, x⊕y = C[x1y1,x2y2,...,xDyD], defned on SD ×SD, and α α α the power transformation, α ⊙ x = C[x1 ,x2 ,...,xD], defned on R × SD, induce a vector space structure on the simplex SD (Pawlowsky-Glahn and Egozcue, 2001). Another important element is the logcontrast, a log-linear combination D D ∑ βi lnxi, with ∑ βi = 0, βi ∈ R (1) i=1 i=1 which plays the typical role of the linear combination of variables (Aitchison, 1986). Once we have a vector space structure, a metric structure is easily defned using the clr scores of a composition x (Aitchison, 1986):   x1 xD clr(x) = ln ,...,ln , g(x) g(x) where g(·) means the geometric mean. Note that the clr(x)k score, being a logcontrast that involves all the parts of a composition, provides information about the relative importance of part xk in the composition. The basic metric elements of the Aitchison geometry as inner product (< ·,· >A), norm (|| · ||A), and distance (dA(·,·)) can be defned as: < x,y >A =< clr(x), clr(y) >E , ||x||2 =< x,x >A , dA(x,y) = ||x⊖ y||A, A where “A” means the Aitchison geometry, “E” means the typical Euclidean geometry, and “⊖” is the perturbation difference x ⊖ y = x ⊕ ((−1) ⊙ y). The metric elements are used to construct orthonormal basis and to calculate the corresponding log-ratio coordinates of a composition (olr(x)) (Egozcue et al., 2003; Mart´ ınFern´ andez, 2019). The expression of these olr-coordinates depends on the selected basis. For example, following Egozcue and Pawlowsky-Glahn (2005) one can defne particular olr-coordinates created through a sequential binary partition (SBP) of a complete composition x = (x1,...,xD). In the frst step of an SBP, when the frst olr-coordinate is created, the complete composition x = (x1,...,xD) is split into two groups of parts: one for the numerator and the other for the denominator. In the following steps, to create the following olr-coordinates, each group is in turn split into two groups. That is, in step k when the olr(x)k-coordinate is created, the rk parts (xn1k ,...,xnrk ) in the frst group are placed in the numerator; the s parts (xd1k ,...,xdsk ) in the second group will appear in the 326 Fundamentals of convex optimization for compositional data denominator; and the rest of D − (rk + sk) parts are not involved in the logratio. As a result, the olr(x)k is r )1/rk rk · sk (xn1k ···xnrk olr(x)k = ln , k = 1,...,D− 1. (2) rk + sk (xd1 ···xdsk )1/sk q rk·sk where is the factor for normalising the coordinate. Note that r1 + s1 = D for rk+sk k = 1. The olr(x)k coordinate, being a logcontrast that involves two groups of parts of a composition, informs, on average, about the relative importance of one group of parts with regard to the other. 3. Convexity on the simplex Following the basic defnitions of convexity on RD (Boyd and Vandenberghe, 2004), the counterpart defnitions of convexity on the simplex SD in a consistent manner with the Aitchison geometry (i.e. A-convexity) are: Defnition 1. Let x1, x2 be two D-part compositions. The A-segment x1x2 is the set x1x2 = {y ∈ SD|y = λ ⊙ x1 ⊕ (1 − λ ) ⊙ x2, λ ∈ [0,1]}. (3) An A-segment can be expressed in olr-coordinates as olr(x1x2) = {z ∈ R(D−1)|z = λ · olr(x1)+(1 − λ ) · olr(x2), λ ∈ [0,1]}. That is the typical expression of a segment of a line on the real space. The defnition of a compositional segment (Eq. (3)) can be used in the defnition of a compositional convex set (A-convex set). Defnition 2. A set B ⊆ SD is an A-convex set if for all x1, x2 ∈ B, the compositional segment x1x2 is contained in B. That is, for any x1, x2 ∈ B and any λ ∈ [0,1], it holds that λ ⊙ x1 ⊕ (1 − λ ) ⊙ x2 ∈ B. To illustrate the defnition of an A-convex set, 3-part compositions were selected in S3 for creating a compositional triangle using the Defnition (1) of an A-segment (Fig.1(a)). By construction, a compositional triangle is an A-convex set. On the upper right, Fig.1(b) shows a typical strip (blue area). A compositional segment (red line) not entirely contained in the set shows the lack of the A-convexity of the strip. Figure 1(b) shows how sets that look like convex sets in the simplex from a typical Euclidean point of view, are not compositional convex sets with the Aitchison geometry, and vice versa (Fig.1(a)). Figures 1(c) and (d), respectively, show the representation of Figures 1 (a) and (b) in olr-coordinates. In these fgures, one recognises the typical form of a triangle and the shape of a non-convex set on the real space. 3.1. Convex hull on the simplex The simplex endowed with the induced Euclidean geometry does not have the same structure and properties as the simplex with the Aitchison geometry. This difference Jordi Saperas Riera, Josep Antoni Mart´ ın Fernandez and Gl` 327 ´ oria Mateu Figueras (a) (b) (c) (d) Figure 1. Triangle and strip in S3: (a) A-triangle (green area); (b) Strip (blue area) with an A-segment (red line); (c) The A-triangle (green area) in olr-coordinates ; and (d) The strip (blue area) with the A-segment (red line) in olr-coordinates. between geometries has implications on the statistical techniques such as the peeling, which is a descriptive statistical technique based on the concept of convex hull (Caussinus, Ettinger and Tomassone, 2012; Small, 1990). Peeling can be described as an iterative algorithm that consists of removing layers of points. Each layer is formed by the points which form the border of the convex hull of the set of remaining points. The convex hull of a set of points {x1,...,xn} in RD is the set n convE (X) = {z ∈ RD|z = λ1x1 + ... + λnxn, ∑ λi = 1, λi ≥ 0,i = 1,...,n}, i=1 This defnition is used by Tolosana-Delgado, von Eynatten and Karius (2011) for CoDa vectors. Among other applications, peeling allows us to graphically represent the centre of a set of points in the last internal layer and can be used for outlier detection (Harsh, Ball and Wei, 2016, chapter 4). We propose the corresponding defnition of the convex hull on the simplex in terms of Aitchison geometry: 328 Fundamentals of convex optimization for compositional data Defnition 3. The A-convex hull of the set of compositions X = {x1,...,xn}∈ SD is n convA(X) = {y ∈ SD|y = λ1 ⊙ x1 ⊕ ... ⊕ λn ⊙ xn, ∑λi = 1, λi ≥ 0,i = 1,...,n}. i=1 Note that the compositional triangle created in Fig. 1(a) is the most simple example of an A-convex hull using the minimum number of compositions. The usefulness of this concept can be illustrated by means of a more complex example in S3. Let X = {x1,...,x20}∈ S3 be a set of compositions randomly generated using a normal distribution on the simplex (Mateu-Figueras, Pawlowsky-Glahn and Egozcue, 2013). Figure 2 compares the A-convex hull convA(X) (on the left) to the Euclidean convex hull convE (X) (on the right). Figure 2(a) shows the successive layers created when peeling is applied to X using the A-convex hull. The last internal layer (i.e., the smallest convex 1 hull) is close to the compositional centre of X, g(X) = 20 ⊙ (⊕20 i=1xi) (green dot), the column-wise geometric mean (e.g., Pawlowsky-Glahn, Egozcue and Tolosana-Delgado, 2015). On the other hand, the typical Euclidean centre of X, the arithmetical mean 1 X = 20 ∑20 i=1 xi (red dot) is far from the last external layer of the peeling, suggesting a potential outlier. Fig. 2(b) shows the layers of the peeling using the E-convex hull. In this case, the Euclidean centre of X (red dot) is inside the last layer, whereas the compositional centre g(X) (green dot) is far from it. In addition, when comparing the frst external layers (i.e., the largest convex hull) in both geometries, one concludes that the A-convex hull fts better with the common arch shape of a normally distributed CoDa set. (a) (b) Figure 2. Peeling applied to a CoDa set X ∈ S3: (a) with A-convex hull; (b) with E-convex hull. Geometric centre g(X) (green dot) and arithmetic centre X (red dot) of CoDa set are plotted. 3.2. Some basic compositional sets The most common sets in convex optimisation in areas such as, among others, design of experiments (DOE), optimisation with mixtures, or time-use data are defned by con- Jordi Saperas Riera, Josep Antoni Mart´ ın Fernandez and Gl` 329 ´ oria Mateu Figueras straining some parts of the composition x: {0 < li ≤ xi ≤ ui < 1; i = 1,...,D} (Chen xi et al., 2010) or by constraining the ratio of two parts: {0 < li j ≤ ≤ ui j; i = j = xj 1,...,D}(Lo Huang and Huang, 2009). 3.2.1. Constraining the ratios between parts: a logcontrast The equation { xi = k ,1 ≤ i = j ≤ D; k > 0} is equivalent to the logcontrast {lnxi − xj lnxj = lnk; 1 ≤ i = j ≤ D}. This equation describes an affne subspace of dimension D − 2 on the simplex with the Aitchison geometry (Egozcue, Pawlowsky-Glahn and xi Gloor, 2018). Consequently, l ≤ xi or ≤ u defne closed half-spaces of the simplex, xj xj and both sets, Π+= {x ∈ SD|l ≤ xi } and Π− = {x ∈ SD| xi ≤ u}, verify the condition of xj xj being an A-convex set (Defnition 2). In general, affne subspaces such as A-lines, A-planes or A-hyperplanes are defned by logcontrasts (Eq. (1)) (Egozcue et al., 2018). A logcontrast splits the simplex into two closed half-spaces, Π+= {x ∈ SD|∑D j=1 βj lnxj ≥ k} and Π− = {x ∈ SD|∑D j=1 βj lnxj ≤ k}. By construction, both half-spaces are A-convex sets (Defnition 2). Figure 3 shows four different logcontrasts (blue lines) whose intersection determines a quadrilateral (green area). Because the intersection of convex sets is a convex set (Boyd and Vandenberghe, 2004) it follows that the quadrilateral is an A-convex set. Figure 3. An A-convex quadrilateral in S3 (green area) determined by the intersection of four half-spaces defned by logcontrasts (blue lines). 3.2.2. Constraining the parts of a composition Despite the fact that an equation such as {xi = k, k ∈ (0,1)}, for any i = 1,...,D, cannot be expressed in terms of a logcontrast, this type of equation also splits the simplex into two sets: the upper set Σ+= {x ∈ SD|xi ≥ k, k ∈ (0,1)}, and the lower set Σ− = {x ∈ α α α α α αα αα α α αα α α α 330 Fundamentals of convex optimization for compositional data SD|xi ≤ k, k ∈ (0,1)}. However, in this case, the two sets are different with regard to their compositional convexity. Proposition 1. For any i = 1,...,D, the set Σ+= {x ∈ SD|xi ≥ k, k ∈ (0, 1)} is an A-convex set. Proof. See Appendix. ■ In contrast, the lower set Σ− is not an A-convex set as is illustrated in Fig. 4(b). Figure 4 shows an example in S3 of the type Σ+ (blue area) and its complementary set, Σ− (grey area), for part x1 with k = 0.4. For each set, two compositions were selected and the corresponding A-segment plotted (red line). Because the red A-segment on Fig. 4(b) is not entirely contained in the set Σ− (grey area), this set is not an A-convex set. (a) (b) Figure 4. A 3-part composition constrained for i = 1 and k = 0.4: (a) The set Σ+ (blue area); (b) The set Σ− (grey area). The red line represents the A-segment for two compositions in each set. The sets Σ+ and Σ− can be generalised as follows: Σ+ i (α) = {x ∈ SD|α1x1 + ... + αDxD ≥ 0,αi > 0,αj ≤ 0, 1 ≤ j = i ≤ D} Σ− i (α) = {x ∈ SD|α1x1 + ... + αDxD ≤ 0,αi > 0,αj ≤ 0, 1 ≤ j = i ≤ D} Note that Σ+ i (α) and Σ− i (α) for α = (−k,...,−k,(1 − k),−k,...,−k), k ∈ (0,1), be- | {z } i come the sets Σ+ and Σ− , respectively. Importantly, an analogous proof to Proposition 1 states that a set Σ+ i (α) is A-convex. On the other hand, Σi −(α) is not an A-convex set. To illustrate these properties, Figure 5 shows the set Σ+ 1 (α) = {x ∈ S4|x1 − 2 3 x2 − 2 3 x3 − 3 1 x4 ≥ 0} (blue area) in S4, which generalises a set of type Σ+= {x ∈ SD|x1 ≥ k, k ∈ (0,1)}. Jordi Saperas Riera, Josep Antoni Mart´ ın Fernandez and Gl` 331 ´ oria Mateu Figueras Figure 5. The A-convex set Σ+ 1 (α) = {x ∈ S4|x1 − 3 2 x2 − 2 3 x3 − 3 1 x4 ≥ 0} (blue area) in S4 . Note that the intersection of sets of type Σ+ or Σ+(α) is an A-convex set. However, when an optimisation problem includes a set of type Σ−(α) the feasible region might be a non A-convex set. In such a case, one should be cautious when applying convex optimisation techniques. 4. Convex functions on the simplex The defnition of a convex function in the Euclidean space can be adapted to the Aitchison geometry following the schema introduced in Luenberger and Ye (2008) and in Boyd and Vandenberghe (2004). Defnition 4. Let W ⊂ SD be an A-convex set. A function f : W → R is an A-convex function if for all x1, x2 ∈ W and λ ∈ [0, 1]: f ((1− λ ) ⊙ x1 ⊕ λ ⊙ x2) ≤ (1− λ ) f (x1)+ λ f (x2) Defnition 5. Let W ⊂ SD be an A-convex set. A function g : W → R is an A-concave function if f = −g is an A-convex function. Note that, as expected, a constant function f (x) = k, with k a real number, is simultaneously an A-convex and A-concave function. Importantly, the classifcation of convex function through the gradient or the Hessian matrix also applies for CoDa. That is, the common rule A twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix of second partial derivatives is positive semidefnite on the interior of the convex set can be used to classify A-convex functions by means of the basic concepts of compositional differential calculus (Barcel´ o-Vidal, Mart´ ın-Fern´ andez and Mateu-Figueras, 2011) working in olrcoordinates. Whereas, using the A-gradient, one has to check the usual expression f (x) ≥ f (y)+ ∇A( f )(y)(ln(x) − ln(y)) 338 Fundamentals of convex optimization for compositional data  2 xj x0 j minimise d2 (x0,x) = ∑3 j=1 ln − ln A g(x) g(x0) (6) x1 ≥ 13 subject to 24 whose standard form is (Example 2)  2 xj x0 j minimise d2 (x0,x) = ∑3 j=1 ln − ln A g(x) g(x0) subject to 13 − 1 ≤ 0 24 ∑3 j=1 xj x1 Figure 9(b) shows that the solution to the A-convex optimisation problem (Eq. 6) �  13 6.87 4.13 is x = . Note that, the Aitchison approach has a reasonable behaviour 24 , 24 , 24 because the largest part of the initial composition (i.e., sedentary time) contributes more to increase the non-sedentary time. The way that the other parts contribute to increase the non-sedentary time is not proportional in any sense. In this case, the (squared) Aitchison distance from x0 to the new centre x is 6.989, smaller than the distance obtained when the new centre is created by perturbation. Importantly, the ratio Sed is not preserved Sleep when moving from x0 to the solution of the optimisation problem (x). That is, using this approach, the parts {Sed, Sleep} are not perturbed by the same factor. (a) (b) 17 6 Figure 9. Time-use optimisation for the composition x0 = ( 1 24 ) (red dot). The dark grey 24 , 24 , = {x ∈ S3|x1 ≥ 13 area is the set Σ+ 24 }. The red diamond x ∈ Σ+ is the closest point to x0. The dashed red line is the contour line for the corresponding distance. (a) Euclidean geometry: x = �  ( 13 11 0 13 6.87 4.13 24 ); (b) Aitchison geometry: x = . Dashed blue line is the perturbation 24 , 24 , 24 , 24 , 24 = ( 13 8.13 2.87 direction from x0 to x 24 ) ∈ Σ+ (blue diamond) 24 , 24 , In the optimisation problem, the movement from x0 to x is explained by the perturbation difference x⊖ x0 = (13/1,6.87/17,4.13/6). Figure 10 shows how the original data set (grey) is moved to the data set perturbed (black) by x⊖x0 = (13/1,6.87/17,4.13/6), preserving the data spread. The centre of the perturbed data set fulfls the condition x1 ≥ 13 24 . With the Aitchison geometry approach, the solution is more realistic. Jordi Saperas Riera, Josep Antoni Mart´ ın Fernandez and Gl` 339 ´ oria Mateu Figueras Figure 10. Resolving the optimisation problem on the ternary diagram: the original data set (grey) is perturbed by x ⊖ x0 = (13/1,6.87/17, 4.13/6) for becoming the new data set (black). The dashed segment is the border of the set x1 ≥ 13 24 . The centre of each data set is shown in red. 6. Conclusions and fnal remarks The adequate defnitions of convex set, convex hull and convex function in a compatible manner with the Aitchison geometry play an important role in the analysis of CoDa. The most popular statistical techniques include optimisation problems that are sensitive to the geometry of the sample space. We have compared the Euclidean geometry to the Aitchison geometry when solving a convex optimisation problem for CoDa. While the Euclidean approach found inconsistent solutions, the Aitchison geometry provided more satisfactory results. This concludes that, despite the fact that any method may give a reasonable solution in some scenarios, it is advisable to use methods that are consistent with the geometry of the sample space. With basic concepts of constrained convex optimisation for CoDa on hand, a revision of some statistical techniques should be carried out. The pending challenge is to investigate the implications in popular techniques such as, among others, outlier detection, experimental design of mixtures, or Lasso regression. 340 Fundamentals of convex optimization for compositional data Appendix: Proofs Proposition 1 For any i = 1,...,D, the set Σ+= {x ∈ SD|xi ≥ k, k ∈ (0,1)} is an A-convex set. Proof. Let x1 = (x11,...,x1D), x2 = (x21,...,x2D) be two D-part compositions in Σ+ . We have to check that for any λ ∈ [0, 1] it holds that (1 − λ ) ⊙ x1 ⊕ λ ⊙ x2 ∈ Σ+ . Due to ∑D j=1 x1 j = ∑D j=1 x2 j = 1 then the equations x1i ≥ k and x2i ≥ k are respectively equivalent to x1i ≥ k(x11 + ... + x1D) and x2i ≥ k(x21 + ... + x2D). Using this equivalence, ∀λ ∈ [0, 1] it holds D D 1−λ λ x x2i ≥ k(x11 + ... + x1D)1−λ (x21 + ... + x2D)λ = k(∑ x1 j)1−λ (∑ x2 j)λ , 1i j=1 j=1 where, with H¨ older’s inequality, the last term of the expression holds D D D 1−λ λ k(∑ x1 j)1−λ (∑ x2 j)λ ≥ k ∑ x1 jx2 j. j=1 j=1 j=1 That is, it holds that D 1−λ λ 1−λ λ x1ix2i ≥ k ∑ x1 jx2 j. (7) j=1 Finally, writing Eq. (7) in terms of the ith part of the composition (1− λ ) ⊙ x1 ⊕ λ ⊙ x2: 1−λλ x x 1i 2i ((1 − λ ) ⊙ x1 ⊕ λ ⊙ x2)i = ≥ k, 1−λλ ∑Dx j=1 x1 j 2 j that is, (1 − λ ) ⊙ x1 ⊕ λ ⊙ x2 ∈ Σ+ . ■ Proposition 2 Let f1 and f2 be two A-convex functions on the A-convex set W ⊂ SD. The function f1 + f2 is A-convex on W. Proof. Let x1, x2 be two D-part compositions in W. For any λ ∈ [0,1] it holds that f1((1 − λ ) ⊙ x1 ⊕ λ ⊙ x2)+ f2((1 − λ ) ⊙ x1 ⊕ λ ⊙ x2) ≤ (1 − λ )[ f1(x1)+ f2(x1)] + λ [ f1(x2)+ f2(x2)]. (8) ■ Jordi Saperas Riera, Josep Antoni Mart´ ın Fernandez and Gl` 341 ´ oria Mateu Figueras Proposition 4 Let f be an A-convex function on the A-convex set W ⊂ SD. The sublevel set G− = {x| x ∈ W, f (x) ≤ α} is an A-convex set for any real number α. α Proof. Let x1, x2 be in G− α and λ ∈ [0, 1], f ((1− λ ) ⊙ x1 ⊕ λ ⊙ x2) ≤ (1− λ ) f (x1)+ λ f (x2) ≤ α. Consequently, (1− λ ) ⊙ x1 ⊕ λ ⊙ x2 ∈ G− α . ■ xi Example 1 The function f (x) = xj , 1 ≤ i, j ≤ D is an A-convex over its domain, dom( f ) = SD . Proof. Let x1, x2 be two D-part compositions and for any λ ∈ [0, 1],  1−λ  λ x1ix2i f ((1− λ ) ⊙ x1 ⊕ λ ⊙ x2) = . x1 jx2 j We apply Young’s inequality, a1−λ bλ ≤ (1 − λ )a + λ b for any a,b ≥ 0 and λ ∈ [0, 1], x1ix2i to the values a = and b = , x1 jx2 j  1−λ  λ x1ix2i f ((1− λ ) ⊙ x1 ⊕ λ ⊙ x2) = ≤ x1 jx2 j x1ix2i (1 − λ ) + λ = (1 − λ ) f (x1)+ λ f (x2). (9) x1 jx2 j ■ Example 2 For any i = 1,...,D, the function f (x) = xi is an A-quasiconcave function over all its domain, dom( f ) = SD . Moreover, using the A-convex function D xj Φα (x) = α ∑ − 1, xi j=1 a superlevel G+= {x ∈ SD| xi ≥ α} for any α ∈ (0, 1) can be represented by means of α Φα (x) ≤ 0. Proof. Through proposition 1, the set G+= {x ∈ SD| xi ≥ α} is A-convex for any real α number α. Because x ∈ SD , the equality ∑D j=1 xj = 1 holds. 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