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Convexity of the Proximal Average 2010
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Title | Convexity of the Proximal Average |
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Johnstone, Jennifer |
Date Created | 2010-07-30 |
Date Issued | 2008 |
Description | The proximal average operator is recognized for its ability to transform two convex functions into another convex function. However, we prove with examples that the proximal average operator does have limitations, with respect to convexity. We also look at the importance of [0; 1] and describe an idea of how to plot the proximal average of two convex functions more efficiently. |
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Language | Eng |
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Computer Science Undergraduate Honours Essays (Okanagan Campus) |
Series | University of British Columbia, Okanagan campus, Computer Science Undergraduate Honours Essays |
Date Available | 2010-07-30 |
DOI | 10.14288/1.0052226 |
Affiliation |
Irving K. Barber School of Arts and Sciences Psychology Computer Science |
Peer Review Status | Unreviewed |
Scholarly Level | Undergraduate |
URI | http://hdl.handle.net/2429/27048 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/42497/items/1.0052226/source |
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Convexity of the Proximal Average by Jennifer Johnstone A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE HONOURS in The I. K. Barber School of Arts & Sciences (Computer Science) THE UNIVERSITY OF BRITISH COLUMBIA (Kelowna, Canada) December, 2008 c© Jennifer Johnstone 2008 Abstract The proximal average operator is recognized for its ability to transform two convex functions into another convex function. However, we prove with examples that the proximal average operator does have limitations, with respect to convexity. We also look at the importance of λ ∈ [0, 1] and describe an idea of how to plot the proximal average of two convex functions more efficiently. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Subdifferential . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Legendre-Fenchel Conjugate . . . . . . . . . . . . . . . . . . 6 2 The Proximal Average . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Convexity Results . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Convexity with respect to x . . . . . . . . . . . . . . 9 2.2.2 Convexity with respect to λ . . . . . . . . . . . . . . 9 2.2.3 Convexity with respect to µ . . . . . . . . . . . . . . 10 2.2.4 Further Investigation . . . . . . . . . . . . . . . . . . 11 3 Plotting the Proximal Average . . . . . . . . . . . . . . . . . 16 3.1 New Plotting function . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 iii Acknowledgements To start, the author would like to express her gratitude to her supervisor Dr. Yves Lucet for his constant support and guidance throughout this process. His ideas and topic suggestions made this thesis possible. The author would also like to thank all those who encouraged her to take on this project and supported her throughout. iv Dedication To my Mother for her continuous encouragement. v Chapter 1 Introduction The purpose of this thesis is to investigate the convexity of the proximal average and elaborate on some of the properties of the proximal average. In this chapter, we review some of the standard facts on Convex Analysis followed by a review of some of the basics of the proximal average in Chapter 2, in which we also provide the main results of our investigation into the convexity of the proximal average. 1.1 Convex Functions In this section we recall some of the basics of Convex Analysis to help the reader better understand the results in Chapter 2. We assume that we are working in a real Hilbert space, H, which is defined as a complete, real, inner product space. The Hilbert space H is complete if all Cauchy sequences in H converge, with respect to the defined norm, and H is an inner product space if an inner product exists such that ‖x‖ = √< x, x >, for all x ∈ H. So we define <· , ·>: H ×H → R such that for all x, y, z ∈ H and λ, µ ∈ R we have < x, y > = < y, x > < λx+ µy, z > = λ < x, z > +µ < y, z > < x, x > ≥ 0 and < x, x >= 0 iff x = 0. Then working in H we can extend any function g : Ω ⊂ H → R to g̃ : H → R ∪ {+∞} with g̃(x) = { g(x) if x ∈ Ω, +∞ if not. We also define dom g̃ as dom g̃ := {x ∈ H|g̃(x) < +∞} 1 1.1. Convex Functions Definition 1.1 (Convex Sets). Let Ω ⊂ H then Ω is a convex set if for all x1 ∈ Ω and x2 ∈ Ω it contains all points αx1 + (1− α)x2, 0 < α < 1. Definition 1.2 (Convex Functions). Let C be a non-empty convex set in H. A function g : C → R is said to be convex on C when, for all pairs (x1, x2) ∈ C × C and all 0 < α < 1, we have g(αx1 + (1− α)x2) ≤ αg(x1) + (1− α)g(x2). For a convex function, g, we see that the line segment {(αx1 + (1− α)x2, αg(x1) + (1− α)g(x2) : α ∈ [0, 1]} is always above the graph of g, as illustrated in Figure 1.1. On the other hand, in Figure 1.2 we see that this is not the case for all pairs (x1, x2) ∈ C × C and all 0 < α < 1. Figure 1.1: For g = x2 we see that the line segment {(αx1 + (1− α)x2, αg(x1) + (1− α)g(x2) : α ∈ [0, 1]} is always above the graph g thus g is a convex function. 2 1.1. Convex Functions Figure 1.2: For g = x4 + 3x3 + 10 we see that the line seg- ment {(αx1 + (1− α)x2, αg(x1) + (1− α)g(x2) : α ∈ [0, 1]}] is not above the graph of g at x = −1 thus g is a non-convex function. We will only consider proper functions, since we are not interested in degenerate functions such as g = +∞ everywhere. Definition 1.3 (Proper). A function g is called proper if g(x) > −∞ for all x and g(x) < +∞ for at least one x. An example of a proper function would be the indicator function which is defined on a nonempty set C by iC(x) = { 0 if x ∈ C, +∞ otherwise. Definition 1.4 (Lower Semi-continuous). A function f is said to be lower semi-continuous (lsc) at a point x̄ ∈ dom f if f(x̄) ≤ lim inf x→x̄ f(x) While continuous functions are lsc, Figure 1.3 gives an example of a lsc function which is not continuous everywhere. 3 1.2. Subdifferential Figure 1.3: An example of a lower semi-continuous function (lsc): it is lsc at x=5 and continuous everywhere else. 1.2 Subdifferential Since convex functions are not always differentiable we need to introduce the concept of a subgradient. Definition 1.5 (Subgradients). Let g : H → R∪{+∞} be a proper convex function and let x ∈ dom g. A vector s in H satisfying g(y) ≥ g(x)+ < s, y − x > ∀y ∈ H is called a subgradient of g at x. Moreover, the set of all subgradients of g at x is called the subdifferential of g at x, denoted by ∂g(x) , see Figure 1.4 for a graph of a function with some subtangent lines. We note that if g is convex and differentiable the subgradient of g at x is 5g(x) such that ∂g(x) = {5g(x)}. An example of when ∂g(x) = {5g(x)} is represented in the tangent of Figure 1.4 when x = 3, as this function is differentiable there. 4 1.2. Subdifferential Figure 1.4: An example of a convex function with both subtangents and tangents. When g is twice continuously differentiable we denote52g(x) its Hessian at x. The Hessian of g is defined as the following symmetric matrix: 52g(x) = ∂2g(x) ∂x21 ∂2g(x) ∂x1∂x2 · · · ∂2g(x)∂x1∂xn ∂2g(x) ∂x2∂x1 ∂2g(x) ∂x22 · · · ∂2g(x)∂x2∂xn · · · · · · · · · · · · ∂2g(x) ∂xn∂x1 ∂2g(x) ∂xn∂x2 · · · ∂2g(x) ∂x2n . It is also worth mentioning that if the Hessian of a function exists then we have the following convexity test. Fact 1.1 (Convexity Test). [7, Theorem 2.69(i)] If we assume that g : H → R is a twice continuously differentiable function then g is convex if and only if its Hessian 52g(x) is positive semi-definite for all x ∈ H. Recall that a Hessian matrix is positive semi-definite if and only if all its eigenvalues are greater or equal to zero [6]. 5 1.3. Legendre-Fenchel Conjugate 1.3 Legendre-Fenchel Conjugate Now that we have defined a convex, lsc and proper function let X = {f : H → R ∪ {+∞} |f is convex, lsc and proper} be the set of functions that we are working with for the remainder of this thesis. In this section we define the Fenchel Conjugate that is important in the field of convex analysis. Definition 1.6 (Legendre-Fenchel Conjugate). The Legendre-Fenchel Con- jugate (aka convex conjugate) of f ∈ X is the function f∗ ∈ X defined by f∗(s) := sup x {< s, x > −f(x)} ∀s ∈ H. Furthermore, we note that the Fenchel Biconjugate Theorem, as seen in [3, 4], states that f ∈ X ⇐⇒ f∗∗ = f. We also define the relative interior of a convex set. Definition 1.7. The relative interior of a convex set C ⊂ H is the interior of C for the topology relative to the affine hull of C. We now present the Fenchel Duality Theorem which allows us to solve convex optimization problems. Theorem 1.1 (Fenchel Duality). [3] Given two functions f and g in X such that y = inf x∈H {f(x) + g(x)} is a finite number and assume the relative interiors of dom f and dom g intersect. Then −y = min x∗∈H [f∗(x∗) + g∗(−x∗)] is actually attained. 6 Chapter 2 The Proximal Average From now on we will assume that f0 and f1 are in X; λ0 and λ1 are two real numbers strictly greater than zero, such that λ0 + λ1 = 1; and µ is strictly greater than zero. 2.1 Main Results We first start by defining the proximal average. Definition 2.1 (Proximal Average). The λ weighted proximal average of f0 and f1 with parameter µ is pµ(f0, f1;λ0, λ1) = 1 µ [ −1 2 ‖x‖2 + inf x1+x2=x [ λ0 ( µf0 ( x0 λ0 ) + 1 2 ∥∥∥∥x0λ0 ∥∥∥∥2 ) +λ1 ( µf1 ( x1 λ1 ) + 1 2 ∥∥∥∥x1λ1 ∥∥∥∥2 )]] .(2.1) Remark 2.1 (Simplification of p). Since λ0+ λ1 = 1 we can re-write equa- tion (2.1) as follows: pµ(f0, f1;λ0, λ1) = pµ(f0, f1; (1− λ1), λ1). Then for λ = λ1 we have pµ(f0, f1;λ) = pµ(f0, f1; (1− λ), λ)). Fact 2.1 (Reformulations). [2, Proposition 4.3] By changing variables we see that Equation (2.1) is equivalent to the following: pµ(f0, f1;λ)(x) = inf (1−λ)x0+λx1=x [(1− λ)f0(x0) + λf1(x1)+ 1 µ ((1− λ)q(x0) + λq(x1)− q(x)) ] . where q(x) = ‖x‖ 2 2 . 7 2.1. Main Results One of the immediate consequences of Definition 2.1, as seen in [2], is that pµ(f0, f1;λ) = µ−1p1(µf0, µf1;λ). This consequence, in conjunction with the definition of the proximal average, provides us with the following two properties: pµ(f0, f1; 0) = f0 and pµ(f0, f1; 1) = f1. The above properties are similarly seen in [2] for µ = 1. As a visualization we see in Figure 2.1 that pµ(f0, f1;λ) is the conversion of f0 into f1 over λ ∈ [0, 1], with constant µ. −15 −10 −5 0 5 10 15 0 50 100 150 200 Proximal Average (Quadratic to Quadratic) x PA 0 0.25 0.5 0.75 1 Figure 2.1: Plot of p1(x2, 2x2;λ), for λ ∈ [0, 1]. We now state some useful and interesting properties that have be previ- ously discovered. Fact 2.2 (Domain). [2, Theorem 4.6] We have the following domain prop- erty dom pµ(f0, f1;λ) = (1− λ) domf0 + λ domf1. Fact 2.3. [3, Proposition 2.8] Let f ∈ X. Then p1(f, f∗; 1 2 ) = 1 2 ‖·‖2 . Fact 2.4 (Fenchel Conjugate of the Proximal Average). [2, Theorem 5.1][3, Fact 2.3] (pµ(f0, f1;λ))∗ = pµ−1(f∗0 , f ∗ 1 ;λ) 8 2.2. Convexity Results 2.2 Convexity Results 2.2.1 Convexity of the Proximal Average with respect to x In this section we will show that the function φ(x) := pµ(f0, f1;λ)(x), with f0, f1, λ and µ fixed, is convex lsc, and proper. Proposition 2.1. [2, Corollary 5.3] Assume that µ > 0, λ ∈ [0, 1] and f0, f1 ∈ X. Then the function φ := x 7→ pµ(f0, f1;λ)(x) is convex for all x ∈ dom p. Proof. Applying Fact 2.4 twice, we see that (pµ(f0, f1;λ))∗∗ = (pµ−1(f∗0 , f ∗ 1 ;λ)) ∗ = p(µ−1)−1(f ∗∗ 0 , f ∗∗ 1 ;λ) = pµ(f0, f1;λ). Hence, by the Fenchel Biconjugate Theorem we have that φ(x) is convex. 2.2.2 Convexity of the Proximal Average with respect to λ In this section we will show that the function φ(λ) := pµ(f0, f1;λ)(x) is convex, with f0, f1, µ and x fixed. In order to show that φ(λ) is convex we need to first recall some properties of marginal and perspective functions. We start with a property pertaining to marginal functions. Fact 2.5. [8, Theorem 2.1.3 (v)] Let Ω, O ⊂ H. If g := Ω×O 7→ R∪{+∞} is convex then the marginal function γ associated to g is convex where γ : O 7→ R, γ(y) := inf x∈X g(x, y) Before we state a useful property of perspective functions we first need to define them. Definition 2.2 (Perspective function). The perspective function of f : Rn → R ∪+∞ is the function from R× Rn to R ∪ {+∞} given by Persp(f)(u, x) = { uf ( x u ) if u > 0 +∞ if not. 9 2.2. Convexity Results Fact 2.6. [5, Proposition IV.2.2.1] [8, Section 1.2] If f : Rn → R∪{+∞} is proper convex, then its perspective function Persp (f) is proper and convex on R× Rn. Now we prove our main result for this section. Proposition 2.2. [1, Proposition 6.1] Assume that the functions f0, f1 are in X, x ∈ dom pµ(f0, f1;λ), and µ > 0. Then the function φ : λ 7→ pµ(f0, f1;λ)(x) is convex for λ ∈ [0, 1]. Proof. Using Definition 2.1 and Remark 2.1, with x1 = x− x0, we have φ : λ 7→ pµ(f0, f1;λ)(x) = inf x0 [g(λ, x0)]− q(x) µ where q(x) = ‖x‖ 2 2 and g(λ, x0) = (1− λ) (µf0 + q) ( x0 1− λ ) + λ (µf1 + q) ( x− x0 λ ) . Now in order to apply Fact 2.5 we need to first show that g is convex as a function of both x0 and λ. We note that φ(λ) = 1 µ min x0 [Persp(µf0 + q) (1− λ, x0) + Persp(µf1 + q)(λ, x− x0)]−q(x) µ based on Definition 2.2. So using Fact 2.6 and [5, Proposition IV.2.1.5] (the composition of a convex function with an affine mapping is convex), the function g is convex. Now, using the convexity of the functions f0 and f1 we have that the function g is bounded below for each value of λ. Hence, Fact 2.5 applies thereby showing that φ is convex. 2.2.3 Convexity of the Proximal Average with respect to µ In this section we will show that the function φ(µ) := pµ(f, λ)(x) is convex, with f0, f1 ∈ X, λ0, λ1 ∈ [0, 1] such that λ0 + λ1 = 1 and x fixed. Proposition 2.3. [1, Proposition 5.7] Assume µ > 0, λ0 + λ1 = 1, λi ≥ 0, fi ∈ X for i = 1, 2 and take x ∈ dom pµ(f0, f1;λ0, λ1) = λ0 dom f0 + λ1 dom f1. Then the function φ : µ 7→ pµ(f0, f1;λ0, λ1)(x) is convex on ]0,+∞[. 10 2.2. Convexity Results Proof. We begin by substituting x1 = (x−λ0x0)/µ into Equation (2.1) and use λ0 + λ1 = 1 to obtain φ(µ) = inf x0 g(µ, x0) where g(µ, x0) = λ0f0(x0) + λ1f1 ( x− λ0x0 λ1 ) + λ0 2µ ‖x0 − x‖2 . Now we see that the function g is lower bounded on any set {µ} × H for any µ > 0, and is convex (as a composition of convex functions). So, Fact 2.5 applies thereby proving that φ is convex. 2.2.4 Further Investigation of the Proximal Average In this section we present our results on the join convexity of the proximal average. We begin with a computation of the proximal average that will be used to build our examples. Lemma 2.1 (Proximal Average of energy functions). [2, Example 4.5] Let f0 = α0q and f1 = α1q where α0 and α1 are strictly positive numbers and q(x) = ‖x‖2 /2. Then pµ(f0, f1;λ) = (( (1− λ) α0 − µ−1 + λ α1 − µ−1 )−1 − µ−1 ) q. Proof. For q(x) = ‖x‖2 /2 in Remark 2.1 we see that using Fact 2.4 and some basic properties of conjugacy, namely (αq)∗ = q∗(α·), we have pµ−1(f0, f1;λ) = ((1− λ)(α0q + µq)∗ + λ(α1q + µq)∗)∗ − µq = ( 1− λ α0 + µ q + λ α1 + µ q )∗ − µq = ( 1− λ α0 + µ q + λ α1 + µ )−1 q − µq. Thus, pµ(f0, f1;λ) = (( 1− λ α0 − µ−1 + λ α1 − µ−1 )−1 − µ−1 ) q. 11 2.2. Convexity Results We now present the main results for this section. Proposition 2.4. The following functions are not always convex: φxλ : (x, λ) 7→ pµ(f0, f1;λ)(x) φxµ : (x, µ) 7→ pµ(f0, f1;λ)(x) φλµ : (λ, µ) 7→ pµ(f0, f1;λ)(x) when f0, f1 are two convex lsc proper functions, λ ∈ [0, 1] and µ > 0. Proof. We need to show that φxλ, φxµ and φλµ are not always convex. So consider the following quadratic example: Let f0 and f1 be in X such that f0 = α0q f1 = α1q where α0 and α1 are strictly positive numbers and q is the quadratic energy function: q(x) = 12x 2 (when x ∈ R). Then using Lemma 2.1 we have pµ(f0, f1;λ) = (( (1− λ) α0 − µ−1 + λ α1 − µ−1 )−1 − µ−1 ) q. To show that φxλ (resp. φxµ and φλµ) is not convex we use Fact 1.1 and show that the Hessian of pµ(f0, f1;λ) is not positive semi-definite, for µ (resp. λ and x) constant. To start, consider pµ(f0, f1;λ) with µ = 1, so as to show that φx,λ(x, λ) is not convex. Now, p1(f0, f1;λ)(x) = 1 2 (( (1− λ) α0 − 1 + λ α1 − 1 )−1 − 1 ) x2. Then with α0 = 2 and α1 = 4 we have φxλ(x, λ) = p1(f0, f1;λ)(x) = 1 2 (( (1− λ) + λ 3 )−1 − 1 ) x2 and the Hessian of φxλ, at x = 2 and λ = 12 , is Hxλ = [ 1 2 3 3 6 ] . 12 2.2. Convexity Results The determinant of Hxλ is −6 which is enough to show that Hxλ is not positive semi-definite, since −6 implies that one of the eigenvalues of the determinant must be negative. Hence φxλ(x, λ) is not convex. Similarly, we can show that φx,µ is not always convex by holding λ constant. So, let λ = 12 then pµ(f0, f1; 1 2 )(x) = 1 2 ( 12 α0 − µ−1 + 1 2 α1 − µ−1 )−1 − µ−1 x2 and when α0 = 2 and α1 = 4 we have φxµ(x, µ) = pµ(f0, f1; 1 2 )(x) = 1 2 ( 12 2− µ−1 + 1 2 4− µ−1 )−1 − µ−1 x2. Now, the Hessian of φxµ, at x = 2 and µ = 1, is Hxµ = [ 1 2 9 2 9 2 −192 ] . The determinant of Hxµ is −25 so φxµ(x, µ) is not convex. Finally, to show that φλµ(λ, µ) is not always convex we consider pµ(f, λ) with f0(x) = 2q(x) and f1(x) = 4q(x), evaluated at x = 2. Then φλµ(λ, µ) = pµ(f0, f1;λ)(2) = 2 (( (1− λ) 2− µ−1 + λ 4− µ−1 )−1 − µ−1 ) and the Hessian of φλµ, at µ = 1 and λ = 12 , is Hλµ = [ 6 1 1 −192 ] . The determinant of Hλµ is −58 so φλµ(λ, µ) is not convex. Altogether, we proved that each of φxλ, φxµ and φλµ are not always convex. Corollary 2.1. The following function is not always convex: φ : (x, λ, µ) 7→ pµ(f, λ)(x) where f = (f0, f1) ∈ X ×X, λ ∈ [0, 1] and µ > 0. 13 2.2. Convexity Results Proof. Corollary 2.1 is a direct result of Proposition 2.1. Proposition 2.5. The following function is not always convex: φ : (f0, f1) 7→ pµ(f0, f1;λ) where f0, f1 ∈ X, λ ∈ [0, 1] and µ > 0. Proof. In order to find a function φ(f0, f1) which is not always convex we need to show, by Definition 1.2, that there exists an x ∈ R such that pµ(τ(f0, f1) + (1− τ)(g0, g1), λ)(x) > τpµ((f0, f1), λ) + (1− τ)pµ((g0, g1), λ)(x) where f0, f1, g0 and g1 are all in X and λ, τ ∈ [0, 1]. So, let f0 = α0q f1 = α1q g0 = β0q g1 = β1q where q is the quadratic energy function q(x) = 12x 2, when x ∈ R. Then for α0 = 1, α1 = 2, β0 = 3 and β1 = 4 the left-hand side is pµ(τ(f0, f1) + (1− τ)(g0, g1), λ) = pµ((τq + (1− τ)3q, τ2q + (1− τ)4q), λ). Then with µ = 5, τ = 12 and λ = 1 2 , we have pµ((τq + (1− τ)3q, τ2q + (1− τ)4q), λ) = p5 ( 2q, 3q, 12 ) = 6527q. On the other hand, the right-hand side is τp((f0, f1), λ) + τp((g0, g1), λ) = τp((q, 2q), λ) + (1− τ)p((3q, 4q), λ) = 1 2 p((q, 2q), 1 2 ) + 1 2 p((3q, 4q), 1 2 ) = 1 2 ( 23 17 q ) + 1 2 ( 127 37 q ) = 1505 629 q. Therefore, pµ(τ(f0, f1) + (1− τ)(g0, g1), λ)(x) > τpµ((f0, f1), λ) + (1− τ)pµ((g0, g1), λ)(x) 14 2.2. Convexity Results since 65 27 ≈ 2.41 > 2.39 ≈ 1505 629 . Hence, φ(f0f1) is not always convex. To conclude this section, we show that extending the proximal average to λ ∈ R does not provide a useful tool as the following example shows the infimum may no longer be attained. Example 2.1. Using Definition 2.1, with Remark 2.1, µ = 1 and x1 = x− x0, we have p1(f0, f1;λ) = −‖x‖ 2 2 + inf x0 [ λ ( f0 ( x0 (1− λ) ) + 1 2 ∥∥∥∥ x0(1− λ) ∥∥∥∥2 ) +λ ( f1 ( x− x0 λ ) + 1 2 ∥∥∥∥x− x0λ ∥∥∥∥2 )]] . (2.2) Then in Equation (2.2) for f0(x) = α0x + β and f1(x) = α1x + β1, with x ∈ R, we have −x 2 2 +inf x0 [ (1− λ) ( α0 x0 (1− λ) + β0 + x20 2(1− λ)2 ) + λ ( α1 x− x0 λ + β1 + (x− x0)2 2λ2 )] which can be further reduced to − x 2 2 + inf x0 [ α0x0 + β0 − λβ0 + x 2 0 2(1− λ) + α1(x− x0) + λβ1 + x2 2λ − x0 λ + x20 2λ ] .(2.3) Then the dominating term of Equation (2.3) is x20 2(1− λ) + x20 2λ) = x20λ+ x 2 0(1− λ) 2λ(1− λ) = x20 2λ(1− λ) which requires 2λ(1− λ) > 0 for the infimum to be finite. Thus we require λ ∈]0, 1[. 15 Chapter 3 Plotting the Proximal Average Given two functions f0 and f1 we want to plot the proximal average of these functions for λ ∈ [0, 1]. 3.1 New Plotting function Currently the Computation Convex Analysis Numerical Library has Scilab functions that can be used to plot the proximal average of two piecewise linear-quadratic (PLQ) functions, namely plq plotpa. A PLQ function is defined on a set of disjoint domains where for each domain the function is either a linear or quadratic polynomial. The plq plotpa function is defined to plot the proximal average curve based on λ, as seen in Figure 3.1. However, plq plotpa is not always very efficient as it may plot the same proximal average curve for a given λ value when the λ values are close together. −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.25 0.5 0.75 1 Figure 3.1: Plot of p1(x2, 2x2 + 1;λ), for λ ∈ [0, 1], done in Scilab using the original command plq plotpa. 16 3.1. New Plotting function So, we created a new plotting function, plot proxavg, that plots each pixel of the proximal average given two PLQ functions, f0 and f1. That is to say for each (x, y) we assign the appropriate λ value where y = p1(f0, f1;λ)(x), (3.1) as seen in Figure 3.2. . −1.105 −0.860 −0.614 −0.368 −0.123 0.123 0.368 0.614 0.860 1.105 −0.105 0.140 0.386 0.632 0.877 1.123 1.368 1.614 1.860 2.105 0 0.25 0.5 0.75 1 Figure 3.2: Plot of p1(x2, 2x2 + 1;λ), for λ ∈ [0, 1], done in Scilab using the new command plot proxavg . The plot proxavg command requires solving Equation (3.1) for λ. This however is not a simple task given the equation of the proximal average, which is why we have taken a different approach. The approach that we have taken assumes f0, f1 are convex and f0 ≤ f1 so that λ is always increasing. Then for every x value we can determine λ starting at y0 = f0(x) for λ = 0 and going to y1 = f1(x) for λ = 1, since p1(f0, f1; 0)(x) = f0(x) and p1(f0, f1; 1)(x) = f1(x). The λ values in between y0 and y1, for each (x, y) where y0 < y < y1, can be determined by incrementally increasing λ until it generates a newy value that is as close to y as possible. 17 Chapter 4 Conclusion We have answered many of the remaining questions pertaining to the prox- imal average including showing that (x, λ, µ, f) 7→ pµ(f, λ)(x) is always convex in x, λ and µ but not always convex in (x, λ), (x, µ), (λ, µ) and (f0, f1). Our plotting algorithm needs further refinement. It cur- rently uses a linear search to compute λ. Implementing a binary search would reduce the computation time. Furthermore, the bounds used in the binary search could be improved by using the fact that the function λ 7→ p(f0, f1;λ)(x) is convex. Future work in this area may focus on defining a partial differential equation equivalent to the proximal average. It is believed that one exists as the proximal average can be described as the curve evolution from f0 to f1 over λ [1]. 18 Bibliography [1] Heinz H. Bauschke, Rafal Goebel, Yves Lucet, and X. Wang. The prox- imal average: applications, extensions and computation. 2008. [2] Heinz H. Bauschke, Rafal Goebel, Yves Lucet, and X. Wang. The prox- imal average: Basic theory. SIAM J. Optim., 2008. [3] Heinz H. Bauschke, Yves Lucet, and Michael Trienis. How to transform one convex function continuously into another. SIAM Rev., 50(1):115– 132, July 2008. [4] Jonathan M. Borwein and Adrian S. Lewis. Convex Analysis and Nonlinear Optimization. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 3. Springer-Verlag, New York, 2000. Theory and examples. [5] Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. Convex Analy- sis and Minimization Algorithms, volume 305–306 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemat- ical Sciences]. Springer-Verlag, Berlin, 1993. Vol I: Fundamentals, Vol II: Advanced theory and bundle methods. [6] D. C. Lay. Linear Algebra and Its Applications. Addison-Wesley Long- man Inc., Reading, Massachusetts, second edition, 2000. [7] A. Ruszczyński. Nonlinear Optimization. Princeton University Press, Princeton, New York, 2006. [8] C. Zălinescu. Convex Analysis in General Vector Spaces. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathe- matical Sciences]. World Scientific, New Jersey, 2002. 19
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