The proximal average for saddle functions and its symmetry properties with respect to partial and saddle conjugacy

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1 The proxmal average for saddle functons and ts symmetry propertes wth respect to partal and saddle conjugacy Rafal Goebel December 3, 2009 Abstract The concept of the proxmal average for convex functons s extended to saddle functons. Selfdualty of the proxmal average s establshed wth respect to partal conjugacy, whch pars a convex functon wth a saddle functon, and saddle functon conjugacy, whch pars a saddle functon wth a saddle functon. 1 Introducton The proxmal average of convex functons s an operaton that produces a proper, lower semcontnuous, and convex functons from a collecton of proper, lower semcontnuous, and convex functons. It was ntroduced n [4] and studed n detal, n a broader framework, n [2]. For more ntuton behnd the concept, see [3]. For an extenson to non-convex but prox-bounded functons, see [6]. For further references, see [2]. The proxmal average avods the ptfalls present when the arthmetc average of convex functons wth dsjont effectve domans s consdered, but does recover the arthmetc average n the lmt, when a certan parameter s drven to 0. A strkng feature of the proxmal average of convex functons s ts self-dualty wth respect to convex conjugacy: the convex conjugate of the proxmal average of a collecton of convex functons s the proxmal average of the convex conjugates of the functons n the collecton. Ths paper extends the concept of the proxmal average from the settng of convex functons to the settng of saddle functons, n a way that preserves the self-dualty. More specfcally, the proxmal average turns out to be self-dual wth respect to partal conjugacy, whch pars a convex functon wth a saddle functon, and wth respect to saddle functon conjugacy, whch pars a saddle functon wth ts conjugate saddle functon. The self-dualty should provde a tool for further study of the proxmal average of saddle functons, by enablng translaton of the propertes of the proxmal average of convex functon va partal conjugacy. 2 Background 2.1 Proxmal average for convex functons A convex functon f : IR n IR, where IR = [, ], s proper f fx for all x IR n and fx < for some x IR n. Gven a proper, lower semcontnuous lsc, and convex f : IR n IR, Department of Mathematcs and Statstcs, Loyola Unversty Chcago, Chcago, IL 60626, USA. Emal: rgoebel1@luc.edu 1

2 ts conjugate functon f : IR n IR s defned by f p = sup x IR n p x fx. The functon f s tself proper, lsc, and convex, and f = f. For more background on convex functons, consult [9]. Consder proper, lsc, and convex functons f 1, f 2,..., f r : IR n IR, numbers λ 1, λ 2,..., λ r > 0 such that r =1 λ = 1, and µ > 0. Let f = f 1, f 2,..., f r, λ = λ 1, λ 2,..., λ r. The λ-weghted proxmal average of f wth parameter µ s P µ λ, f = λ 1 f 1 + µ j λ 2 f 2 + µ j... λ r f r + µ j µj, 1 where s the operaton of ep-multplcaton,.e., gven α > 0, φ : IR n IR, α φx = αφx/α for all x IR n ; s the operaton of ep-addton or -convoluton,.e., gven φ 1, φ 2,..., φ r : IR n IR, φ 1 φ 2... rx = φ 1 x 1 + φ 2 x φ r x r x 1 + x x r = x for all x IR n ; and jx = 1 2 x 2 for all x IR n. For more background on the operatons of epmultplcaton and -convoluton, consult [11]. Regardng the proxmal average, Theorem 5.1 and Corollary 5.2 n [2] state ths: Theorem 2.1 For proper, lsc, and convex functons f 1, f 2,..., f r : IR n IR, numbers λ 1, λ 2,..., λ r > 0 such that r =1 λ = 1, and µ > 0, the functon P µ λ, f s a proper, lsc, and convex functon. Furthermore, P µ λ, f = P µ 1λ, f, where f = f 1, f 2,..., f r. Further propertes, for example the convergence of P µ λ, f to the arthmetcal average of functons f when µ 0 and to the ep-graphcal average when µ, can be found n [2]. 2.2 Saddle functons In ths artcle, a saddle functon s a functon h : IR m IR n IR such that hx, y s convex n x for each fxed y and concave n y for each fxed x. A saddle functon h s proper and closed f ts convex parent f : IR m IR n IR and ts concave parent g : IR m IR n IR, obtaned from h va partal conjugacy formulas fx, q = sup y IR n hx, y + q, y, gp, y = x IR m hx, y p, x 2 are such that f and g are proper convex functons conjugate to each other, that s, gp, y = sup p, x + y, q fx, q. 3 x IR m,q IR n An alternatve defnton of a closed saddle functon, one that reles on lower and upper semcontnuous closures, can be found n [11, Secton 34]. The equvalence class [h] contanng a proper and closed saddle functon h conssts of all proper and closed saddle functons that have the same 2

3 parents as h. Ths equvalence class can be also descrbed as the set of all saddle functons h such that h h h, where the lowest element h and the hghest elements h of the class are gven, respectvely, by hx, y = sup p IR m gp, y + x, p, hx, y = q IR n fx, q y, q. 4 Each of the two formulas above yelds a proper and closed saddle functon for every proper and upper semcontnuous functon g and every proper and lower semcontnuous functon f, respectvely. For a proper and closed saddle functon h, ts doman dom h s the product set C D, where C = x IR m hx, y < y IR n, D = y IR n hx, y > x IR m. Proper and closed saddle functons from the same equvalence class have the same domans. Gven an equvalence class [h] of proper and closed saddle functons, the class conjugate to t n the saddle sense, denoted [h ], has the lowest and the greatest elements gven by h p, q = sup x IR m h p, q = y IR n p, x + q, y hx, y, y IR n sup p, x + q, y hx, y, x IR m 5 where h s any functon n [h]. In other words, the class conjugate to [h] comes from a convex parent p, y gp, y and a concave parent x, q fx, q. For detals, see [8], [9], and for the nte-dmensonal case, [10] and [1]. For a smple example of a saddle functon, consder hx, y = ajx bjy for a, b > 0. The equvalence class [h] conssts of a sngle element h. Then the convex and concave parents of h are gven, respectvely, by fx, q = ajx+b 1 jq and gp, y = a 1 jp bjy. The class conjugate to h conssts of a sngle element as well, gven by h p, q = a 1 jx b 1 jq. All the calculatons follow smply from the fact that, for convex conjugacy, aj = a 1 j. In partcular, h = h f a = b = 1. A more nterestng example of a saddle functon s gven by hx, y = x y, n the case of m = n. The convex and concave parents of h are gven, respectvely, by fx, q = δ 0 x + q and gp, y = δ 0 y p, where δ 0 : IR n IR s gven by δ 0 0 = 0, δ 0 x = f x 0. The saddle conjugate of h s gven by h p, q = p q, so h = h. The operaton that generalzes the ep-addton, or -convoluton, of convex functons to the settng of saddle functons s that of extremal convoluton. Followng [7], we say that the extremal convoluton of [h 1 ], [h 2 ],..., [h r ] s well-defned f all the convex-concave functons of the form x, y sup P y=y,y D h x P, y, 6 x=x,x IR m and of the form x, y P x=x,x C sup h x P, y 7 y=y,y IR n where h [h ] and C D = dom h, = 1, 2,..., r, belong to a sngle equvalence class. That equvalence class s then denoted [h 1 h 2... h r ]. A suffcent condton for the extremal convoluton of [h 1 ], [h 2 ],..., [h r ] to be a well-defned class of proper and closed saddle functons s that [h 1], [h 2],..., [h r] be fnte-valued; see [7, Theorem 2]. In such a case, replacng h by h n 6 and h by h n 7 yelds, respectvely, the least and the greatest elements of [h 1 h 2... h r ]; see [7, Theorem 3]. 3

4 3 Proxmal average for saddle functons Throughout ths secton, let [h 1 ], [h 2 ],..., [h r ] be equvalence classes of proper and closed saddle functons; let λ 1, λ 2,..., λ r > 0 be such that r =1 λ = 1; and let µ, η > 0. For notatonal convenence, we wll let j x, j y : IR m IR n IR be gven by j x x, y = jx and j y x, y = jy. We wll also wrte h for [h 1 ], [h 2 ],..., [h r ]. 3.1 Defnton and man results Extendng the proxmal average to the settng of saddle functons requres replacng the -convoluton n 1 by the extremal convoluton. In 1, the convex functons are augmented by a quadratc term before the -convoluton s taken. A smlar procedure, n the saddle settng, ensures that the extremal convoluton s well-defned. Lemma 3.1 Let h = h + µ j x η j y, = 1, 2,..., r. Then the extremal convoluton ] [λ 1 h 1 λ 2 h 2... λ r h r s well-defned. Proof. For = 1, 2,..., r, gven any choce of h [h ], the functon h = h + µ j x η j y s strongly convex n x, strongly concave s y, and hence h s fnte-valued; see for example [9, Theorem 37.3]. The same conclusons apply to λ h. Now, [7, Theorem 2] fnshes the proof. The proxmal average of [h 1 ], [h 2 ],..., [h r ], wth coeffcents λ 1, λ 2,..., λ r and parameters µ, η > 0, s well-defned f all convex-concave functons on IR m IR n of the form ] h [λ 1 h 1 λ 2 h 2... λ r h r, h µ j x + η j y, where 8 h = h + µ j x η j y, h [h ], = 1, 2,..., r, belong to a sngle equvalence class. In such a case, the proxmal average of [h 1 ], [h 2 ],..., [h r ] wll be represented as [ P µ,ηλ, h ] ] = [λ 1 h 1 λ 2 h 2... λ r h r µ j x + η j y. Theorem 3.2 The proxmal average of [h 1 ], [h 2 ],..., [h r ], wth coeffcents λ 1, λ 2,..., λ r and parameters µ, η > 0, s well-defned. Theorem 3.2 s proved, along wth Theorem 3.3, n the next secton, by showng that the least and the greatest saddle functons of the form 8 are obtaned va partal conjugacy, as n 4, from a par of functons, one convex and one concave, related to one another as n 3. The general theory of saddle functons then ensures that all saddle functons of the form 8 are equvalent. In fact, the par of proper, lsc, and convex functons mentoned above, the concave and convex parents of [ P µ,ηλ, h ], turn out to be approprately understood proxmal averages of the convex and the concave parents of [h 1 ], [h 2 ],..., [h r ]. More precsely, let f and g be, respectvely, the convex and the concave parent of [h ], = 1, 2,..., r. Let Pµ,η λ, f be the proxmal average of the convex 1 functons f 1, f 2,..., f r wth coeffcents λ 1, λ 2,..., λ r and parameters µ, η 1, defned by P µ,η 1λ, f = λ 1 f 1 λ 2 f 2... λ r f r µ j x 1 η j y 9 4

5 where f = f +µ j x + 1 η j y, = 1, 2,..., r. Let Pµ 1,ηλ, g be the proxmal average of the concave functons g 1, g 2,..., g r wth coeffcents λ 1, λ 2,..., λ r and parameters µ 1, η, defned by Pµ 1,η λ, g = λ 1 ĝ 1 λ 2 ĝ 2... λ r ĝ r 1 µ j x η j y where ĝ = g 1 µ j x η j y, = 1, 2,..., r. Theorem 3.3 The convex and the concave parents of the proxmal average of [ Pµ,ηλ, h ] are, respectvely, the functons Pµ,η λ, f and P 1 µ 1,ηλ, g. A straghtforward consequence of Theorem 3.3 s that the conjugate functon, n the saddle sense, to the proxmal average of [h 1 ], [h 2 ],..., [h r ], s the proxmal average of [h 1], [h 2],..., [h r]. Corollary 3.4 The equvalence classes [ Pµ,ηλ, h ] [ ] and Pµ 1,η λ, h are conjugate to one another n the sense of saddle functon 1 conjugacy. Proof. Theorem 3.3 says that the concave parent of [ Pµ,ηλ, h ] s Pµ 1,ηλ, g. The convex parent of the class conjugate to [ Pµ,ηλ, h ] s the functon p, y Pµ 1,ηλ, gp, y; recall 5 and the comments followng t. Now, Pµ 1,ηλ, gp, y = P µ 1,η λ, g p, y where g p, y = gp, y and [ the functons p, y g p, y are convex parents of [h ]. Hence the class conjugate to P µ,ηλ, h ] has, as ts convex parents, the proxmal average of convex parents of [h ]. Another use of Theorem 3.3 shows that the class conjugate to [ Pµ,ηλ, h ] [ ] s exactly Pµ 1,η λ, h. 1 When, for every = 1, 2,..., r, h = a j x b j y for some a, b > 0, the proxmal average can be found explctly, thanks to the formula for the convex proxmal average of convex quadratc functons; see [2, Example 4.5]. Here, as an llustraton, we fnd the proxmal average of the saddle functons h 1, h 2 on IR n IR n gven by: wth parameters µ = η = 1. One has: h 1 x, y = 1 2 x y 2, h 2 x, y = x y, h1 x, y = x 2 y 2, h2 = x y x y 2, λ 1 h 1 x, y = 1 λ 1 x 2 1 λ 1 y 2, λ 2 h 2 x, y = 1 λ 2 x y + 1 2λ 2 x 2 1 2λ 2 y 2, and the proxmal average P1,1 λ, hx, y s gven by x 1+x 2=x 1 sup x x x 2 y 2 1 y y 2 2 y 1+y 2=y λ 1 2λ 2 λ 2 λ 1 2λ x y 2. The computaton nvolves mmnzng and maxmzng lnear-quadratc functons and s straghtforward. One obtans P1,1 λ, hx, y = 1 λ λ x y 2 + 2λ λ 2 x y. 2 5

6 Note that P1,1 λ, h = h 1 when λ 1 = 1 and λ 2 = 0, whle P1,1 λ, h = h 2 when λ 2 = 1. An nterestng feature of P1,1 λ, h s ts self-dualty wth respect to saddle conjugacy: P1,1 λ, h = P1,1 λ, h. Ths self-dualty follows from h 1 = h 1, h 2 = h 2, and Corollary 3.4 but can be also verfed drectly. For the case of n = 1, the self-dualty of such a functon was noted n [5]. A geometrc nterpretaton of P1,1 λ, h, for the case of n = 1, s that ts graph s obtaned from the graph of h 1 through the rotaton, about the lne x = y = 0, by the angle α such that sn 2α = 2λ 2 /1 + λ 2 2. When λ 2 = 1 and α = π/4, one obtans the graph of h 2. Theorem 3.3 suggests an alternatve way to compute P 1,1 λ, h. The convex parents of h 1, h 2 are gven by, respectvely, f 1 x, q = 1 2 x q 2, f 2 x, q = δ 0 x + q. The proxmal average of f 1, f 2, P1,1 λ, f s gven at x, q by 1 x q δ 0 x 2 + q x q x 1+x 2=x,q 1+q 2=q λ 1 λ 1 2λ 2 λ 1 2 x q 2, whch smplfes to 1 x x q + x x x 2 λ 1 λ 1 λ 2 2 x q 2. A smpler than n the saddle case computaton yelds P1,1 λ, fx, q = 1 + λ λ x 2 2λ 2 1 λ 2 x q λ λ q 2. Then, the relatonshp P1,1 λ, hx, y = q P 1,1 λ, fx, q y q yelds the same formula for P1,1 λ, h as obtaned before. 3.2 A key proposton Let H : IR m IR n IR be gven, at each x, y IR m IR n, by where h = h + µ j x η j y, = 1, 2,..., r, and hx, y = Hx, y = hx, y µ jx + η j y P x=x,x λ C Proposton 3.5 For all x, y IR m IR n, sup λ P h x, y. y=y,y IR n P q IR n µ,η 1λ, fx, q y q = Hx, y. 10 The remander of ths secton s devoted to provng ths result. The summaton symbol n what follows always means r =1. The left hand sde of 10 equals q P x: x=x P q: q=q H 1 q, q, x, where q = q 1, q 2,..., q r, x = x 1, x 2,..., x r and H 1 q, q, x = [ x λ f, q + 1 λ λ λ µ jx + η ] jq 1 jx ηjq y q. λ µ 6

7 Wth no loss of generalty, the mum can be taken over x such that, for = 1, 2,..., r, x s such that there exsts q wth dom f, and thus, over x λ C, where C D = dom h. x λ, q λ Ths fact, swtchng the order of takng the ma, and then usng the frst equaton n 2, wth the greatest element h of [h] n place of h, and wth y /λ takng place of y, turns the left hand sde of 10 to x: P x=x,x λ C q q: P q=q sup y H 2 q, x, y, where y = y 1, y 2,..., y r and H 2 q, q, x, y = [ x λ h, y + η jq + y ] q λ λ λ λ ηjq y q + 1 λ µ jx 1 µ jx. x As x λ C, the functon h λ, s upper semcontnuous see the dscusson above [9, Theorem 33.3] and proper as a concave functon, n the sense that t does not take on the value and s not dentcally see [9, Theorem 43.3]. Thus, the frst lne n the formula for H 2 above s a proper and closed saddle functon of q, y IR m r IR n r. It s also strongly convex n q and, consequently, t has no drectons of recesson n q; see [11, Theorem 8.7]. Then, [9, Theorem 37.3] ensures that swtchng the order of the mum and supremum s possble. Thus, the left hand sde of 10 becomes P x: x=x,x λ C q sup y P q: q=q H 2 q, q, x, y. Some smplfcaton s now possble, thanks to the followng lemma. Lemma 3.6 [ η q: P jq + y q q =q λ λ ] = 1η j y + ηjq + q y 1 ηλ jy 11 Proof. Let φ : IR n IR be gven by φ q = η λ jq + y λ q, = 1, 2,..., r; let Φ : IR nr IR be the convex and fnte-valued functon gven by Φq = φ q ; let A : IR nr IR n be gven by Aq = q ; and let Ψ : IR n IR be the ndcator functon of q,.e., Φy = 0 f y = q, Φy = f y q, whch makes Ψ proper, lsc, and convex. The left hand sde of 11 s Φq + ΨAq. q IR rn Snce Φ s fnte-valued, Fenchel Dualty see, for example, [9, Corollary ] yelds that the mum above equals sup Ψ v Φ A v = sup q v φ v = sup q v φ v. v IR n v IR n v IR n Now, φ v = sup q IR n = η λ j v q y λ η v y λ q η jq λ λ = λ η j = η λ sup v y λ q IR n λ η v y q jq λ 7

8 and sup q v φ v v IR n = sup v IR n = sup v IR n q v λ η j v y λ q v 1η jv + 1η v y 1 η 1 λ jy = 1 η sup y + ηq v IR n = 1 η j y + ηq 1 1 jy η λ v jv 1 1 jy η λ whch expands to the rght hand sde of 11. Lemma 3.6 shows that the left hand sde of 10 s P x: x=x,x λ C q sup y H 3 q, x, y, where H 3 q, x, y = q y y + 1 η j y + [ x λ h, y 1 ] jy λ λ ηλ 1 µ jx + 1 λ µ jx. Lemma 3.7 below shows that, for every x λ C, where q sup H 3 q, x, y = y sup y: P H 4 x, y y =y H 4 x, y = [ x λ h, y + 1 λ λ λ µ jx 1 ] jy 1 ηλ µ jx + 1 η j y. Consequently, the left hand sde of 10 s Ths verfes 10. x: P sup x =x,x λ C y: P H 4 x, y. y =y Lemma 3.7 Gven x λ C, let φ : IR n r IR be gven by φy = 1 η j y + [ x λ h, y 1 ] jy. λ λ ηλ Then Proof. The functon q y λ j sup q y y φy = y y: P φy. y =y y λ j y = λ j y λ j y λ s convex, fnte-valued, and hence contnuous. Indeed, t s quadratc and, by convexty of j, nonnegatve. Furthermore, t s strongly convex n every drecton except y 1 /λ 1 = y 2 /λ 2 = = y r /λ r. Consequently, the functon φ s convex, proper, lower semcontnuous, and q sup y q y y φy = q φ A T q q y = sup y q φ A T q, q 8 λ

9 where A : IR n r IR n s gven by Ay = y, and so A T q = q, q,..., q IR n r. The statement of the lemma amounts to ψ y = y: P y=y φy for the convex functon ψ : IR n IR gven by ψq = φ A T q. Ths holds, by [11, Theorem 11.23], f 0 nt dom φ rgea T. Suppose that ths constrant qualfcaton fals: there exst v = v 1, v 2,..., v r IR n r, v 0, such that v w v A T q 0 for all w dom φ, q IR n. In partcular, v w 0 for all w dom φ whle 0 = v A T q for all q IR n. Snce φ s strongly convex n every drecton except possbly y 1 /λ 1 = y 2 /λ 2 = = y r /λ r, the horzon functon φ of φ s such that φ y = except possbly when y 1 /λ 1 = y 2 /λ 2 = = y r /λ r. By [11, Theorem 11.5], ths horzon functon s the support functon of dom φ. Thus, v w 0 for all w dom φ mples that v 1 /λ 1 = v 2 /λ 2 = = v r /λ r. But then 0 = Av q = 1 + λ 2 /λ λ r /λ 1 v 1 q for all q IR n mples v = 0. Ths s a contradcton. 3.3 Proof of Theorems 3.2 and 3.3 Snce the functons h n the defnton of H are the greatest elements of [ h ], = 1, 2,..., r, the functon H s the greatest functon of the form 8. The functon H turns out to be a partal conjugate, as n the second formula n 4, of a proper, lsc, and convex functon Pµ,η λ, f. Thus 1 H s a proper and closed saddle functon, and n partcular, there exsts a proper and closed saddle functon of the form 8. Let H : IR m IR n IR be gven, at each x, y IR m IR n, by where h = h + µ j x η j y, = 1, 2,..., r, and hx, y = Hx, y = hx, y µ jx + η j y sup P y=y,y D h x P, y. x=x,x IR m Snce h are the least elements of [ h ], = 1, 2,..., r, the functon H s the least functon of the form 8. The functon H turns out to be a partal conjugate, as n the frst formula n 4, of a proper, lsc, and convex functon Pµ 1,ηλ, g. Indeed, Arguments symmetrc to those showng Proposton 3.5 mply the followng: for all x, y IR m IR n, sup Pµ 1,η λ, gp, y x p = Hx, y. 12 p IR m Thus H s a proper and closed saddle functon. The proper and lower semcontnuous convex functons Pµ,η λ, f and P 1 µ 1,ηλ, g turn out to be conjugate to one another. Ths s a consequence of f and g beng conjugate to one another, = 1, 2,..., r, and the self-dualty of the proxmal average n the convex settng, as summarzed below. Lemma 3.8 For all p, y IR m IR n, Pµ,η 1λ, f p, y := sup p x + y q Pµ,η 1λ, fx, q = Pµ 1,η λ, f p, y. x,q IR m IR n Proof. For the functons f, = 1, 2,..., r n 9, one has, for all x, q IR m IR n, f x, q = f x, q + µ j x x, q + 1 η j yx, q = f x, q + µ jx + 1 η jq 9

10 µ x 1 x = f µ, ηq + j µ + j ηq η = g Ax, q + j Ax, q x where Ax, q = µ, µx, 1 µq and g x, q = f η q = f A 1 x, q. Thus, for all x, q IR m IR n, Pµ,η 1λ, fx, q = P 1,1 λ, g Ax, q. Self-dualty of the proxmal average for convex functons, as stated n Theorem 2.1, and [9, Theorem 12.3] combned wth the fact that the lnear mappng A s nvertble and self-adjont, Pµ,η 1λ, f p, y = P 1,1 λ, g A 1 p, y = P1,1 λ, g A 1 p, y. [9, Theorem 12.3] also gves that g p, y = f Ap, y. A calculaton, smlar to the one carred out above for f, fnshes the argument. Conjugacy of the functons Pµ,η λ, f and P 1 µ 1,ηλ, g, and ther relatonshp to H and H captured by 10 and 12 mples that all saddle functons H such that H H H are proper, closed, and form an equvalence class. Snce H and H are, respectvely, the least and the greatest saddle functons of the form 8, the proof of Theorem 3.2 and Theorem 3.3 s complete. References [1] H. Attouch, D. Azé, and R. J-B Wets. Convergence of convex-concave saddle functons: contnuty propertes of the Legendre-Fenchel transform and applcatons to convex programmng and mechancs. Ann. Inst. H. Poncaré Anal. Non Lnéare, 5: , [2] H. Bauschke, R. Goebel, Y. Lucet,, and X. Wang. The proxmal average: basc theory. SIAM J. Optm., 192: , [3] H.H. Bauschke, Y. Lucet, and M. Trens. How to transform one convex functon contnuously nto another. SIAM Rev., 50: , [4] H.H. Bauschke, E. Matoušková, and S. Rech. Projecton and proxmal pont methods: convergence results and counterexamples. Nonlnear Anal., 565: , [5] R. Goebel. Convexty, Convergence and Feedback n Optmal Control. PhD thess, Unversty of Washngton, Seattle, [6] W. L. Hare. A proxmal average for nonconvex functons: a proxmal stablty perspectve. SIAM J. Optm., 202: , [7] L. McLnden. Dual operatons on saddle functons. Trans. Amer. Math. Soc., 179: , [8] R.T. Rockafellar. A general correspondence between dual mnmax problems and convex programs. Pacfc J. Math., 25: , [9] R.T. Rockafellar. Convex Analyss. Prnceton Unversty Press, [10] R.T. Rockafellar. Conjugate Dualty and Optmzaton. Number 16 n CBMS-NSF Regonal Conference Seres n Appled Mathematcs. SIAM,

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