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Polyhedral Approximations in Convex Optimization

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Polyhedral Approximations in Convex Optimization Polyhedral Approximation Extended Monotropic Programming Special Cases Polyhedral Approximations in Convex Optimization Dimitri P. Bertsekas Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Tsinghua University, 2009 Polyhedral Approximation Extended Monotropic Programming Special Cases Convex Optimization Algorithms: Generalities Primary methodology for large scale problems. Arise in the context of duality, network optimization, machine learning. Principal methods date to the 60s ... but have been used in new ways recently. Descent methods (e.g., subgradient). Polyhedral approximation (e.g., cutting plane). Proximal (regularization) methods - possibly in combination with polyhedral approximation. Polyhedral Approximation4 Extended Monotropic Programming Special Cases Our Focus in this Talk 4 A unifying framework for polyhedral approximation methods. IncludesPo classicallyhed methods:ral Convexity Template Cutting plane/Outerx h2(λ) linearization− h1(λ) h2(λ∗) − h1(λ∗) x∗ (−λ∗, 1) (−λ, 1) Slope = λ SimplicialS decomposition/Innerlope = λ∗ linearization x h2(λ) − h1(λ) h2(λ∗) − h1(λ∗) x∗ (−λ∗, 1) (−λ, 1) Slope = λ Slope = λ∗ IncludesPoly newhed methods,ral C ando newnv versions/extensionsefx1(xi) tfy2(x) Tem ofp oldla methods.te Based onf a1(x convex)(a)f2(bx) conjugacy framework and outer/inner linearization (a) (b) duality. minimize f(x) = x1 + x2 minimize f(x) = x1 + x2 subject to g(x) = x1 ≤ 0 subject to g(x) = x1 ≤ 0 x ∈ X = (x1, x2) | x1 > 0} x ∈ X = (x1, x2) | x1 > 0} Outer linearization! of f ! f ∗ = ∞, q∗ = −∞ ∗ x0 x∗ 1 x2 x3 x4 f(x) X x f = ∞, q = −∞ F (x) H(y) y h(y) ! f(x0S) =+ (x(x−2,xx0))|g0x > 0 S = (x2, x) | x > 0 ! " ! " sup inf φ(x, zOu) tersuplineari!inf 2zφˆation(x, z)of=fq = p˜(0) p(0) = w = inf sup φ(x, z) 2 fq((xµ1))=+m(xin−xx1+)µgx1 ∗ ∗ q(µ) = min x + µx z Z x X x∈≤# z Z x X ≤ x X z Z x∈# ∈ ∈ ∈ ∈ ∈ ∈ ! " ! " xx00−xx101/(xx412µ2x)gx203gigf01xµg4g12>xfg00(2x1) xX2Slgopx0 geSl1=Slopg2opye0 e=Sl=−opy10y/e0Sl(=4Slopµopy)e1 e=Slif=opyµ1ye>1Sl=Sl0opopye2Fe=(x=y)2yH2 (y) y h(y) n = Shapley-Fαx + (1 −=olαk)my,an0 ≤Theorem:α ≤ 1 Let S = S1 + + Sm with Si , # −∞ if µ ≤ 0 i =Ou1,te. .rOu. ,OuLimtenearizationter#rLi−Linearization∞nearizationof fif µofof≤ff0 · · · ⊂ # Slope = y0 Slope = y1 Slope = y2 ! If sf(xcon0) v+(S()xthen− x0s)=g0s1 + + sm where minimiOuzePte(Purf)(Li(u=x)nearization)c==mc−amxx{a0x,{Pu0}(,ofuu)}f= c m∈maxin{i0m, uiz}e f◦(x) = −Fx(x·)·FH·F((x(y)x))HyH(hy(()yy)y)yh(hy()y) Slope = y0 Slope = y1 Slope =s y2 conv(S <) for90 all i = 1, . , m, ˆ i sup iinf φ(x, z) sup inf φ(x, z) = q∗ = p˜(0) p(0) = w∗ = inf sup φ(x, z) subjeSlctoptoe =gy(x0)Sl=opxe−=1/y21 ≤Slop0 e =∈ syu2bject to g(x) =≤x − 1/2 ≤ 0 ≤ Outer Linearizationc c of f si c Si∗ zforZ atx 2Xleast m nz Z1xindX ices i. x X z Z Outer LinearizationLevel set !xof|ffF(x(x))≤∈Hf(y)+∈yαhc∈(y/)2" Opt−!ima−∈l sol∈ution set x0 ∈ ∈ Ouxte∈rXLi=nearizationOu{t0e,r1A} pprofoxifmf(axtThei1o)n+osuf(mfxIof−nxnaxe∈r1largeX)Lign=1ena{umr0iz, a1bt}erionofofconCovnexjugsetsate ish almost convex n ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Shapley-F∗ olkman Theorem: Let S = S1 + + Sm with Si , x xh(λh)(λh) 1(hλ1)(λh)x2(hλh2()λ−))h−1(hλ1)()λhFx)2k((λxx()F)−H−(λx−kh),(λ1λH(),λ(1y()))−y(λx−Non,h1λ(),y(1−)c)onλ v,exit1) y(−ofλ,the1ˆ) sumˆˆis caused by a small number (n + 1) of sets q(µ) = min −x + µ(x − 1F/sup2()x) inHsupfsup(yφ)(inyxinf,hzfφ()y(φx)(X,xzsup,)z) insupfsupφ(inxinf, zfφ)(φ=x(,xqz,∗)z)==qp˜∗(0)q∗==p˜(0)p˜p(0)(0) =p(0)pw(0)∗==·=·w·infw∗ ∗==supinfinfφsup(xsup, zφ)(φx(,x⊂z,)z#) x∈{0,1} z Z xFzX(qzxZ(µ)Zx)Hix=X=(Xy1)m,≤y.i.nzh. (,Zym≤)−x≤zxXz+ZZxµx(XxX− 1/2) ≤ ≤≤ x X zx ZxXXz zZZ ! c c ∈" ∈∈c∈ ∈∈ x∈{0,1∈} ∈∈∈ ∈∈ ∈ ∈∈∈ ∈∈ p(v) + "v"2 2ˆp(vα)x++ "(v1If"−2s α)cony, v0(S≤) αthen≤ 1s =˜ s1 + ˇ + sm where sup=inmfinφ{(−x,µzp/)(2v,)µ+sup/2 −"inv1"f} φ(x, z) = q∗ = p˜x∈(0)0 x1 x!p2(0)g0 =g1wg∗2 =f(infx) =sup"·(·cl·φ)(fx(,xz)) z Z x X ≤ 2z Z2 x X 2 s= mconin{≤v−(µS/)2,forµ/2al−l i1=}x 1X, .z. Z, m, n n n ∈ ∈ ∈ ∈ Shapley-FShapley-FShapley-Foli xkkm−olanolαkkmTheorem:gmikananTheorem:Theorem:Let∈ SLet∈Let= SSS1=+=S1S1+++Sm++withSmSmwithSwithi SiSi , , , ∗ ˆ ∈ · · · · ···· · ⊂ # ⊂⊂## (0, f ) = (0,sup0) in(−f 1φ/(2x,,0z)) (1/sup2, −in1)f φ(x, z)i==q1∗, =.i. =.p˜,(0)ms1, . .Sp, m(0)for=atwl∗east= infm supn φ(1x,indz)ices i. ∗ ˆ i = 1, .i. , mi z Zsupx Xinf φ(x,≤z)z Zsupx (X0in, ff )φ(=x,(z0)ˆ, 0=) q∗(−=1p˜/∈(0)2≤, 0) (p◦1(0)/2,=−w1)∗x=X−infzˇ Zsup− φ(x, zn) −1/2 1 ∈ ∈Shapley-Fsup infiolnφfk(i∈nxmLf,czan(L)∈xc,(λxTheorem:ksup, )λk)iniIffnfsφL(xcLet,xcon(Ifzxk),+sλv=1Sk(=S)qPcon=∗)P(then=uXSThev)1p(˜(=S(0)x+)kscsuthen−=mmaαsp+xofk(0){sg+qS0ka=,m)=u∈=larges}withw(+cl∗+∈s=)npumS(0)infwherei+bsersuppof(0)whereφ,con(x=,vzwex) sets is almost convex z Z x X ≤ z Z x X x X If s con<v(9≤S0) then· · · s1 =∗ s1 x1+ Xmz +Z s⊂mm#where ∗ ∈i =∈1z, .Z.x. , mXx∈Xx∈X ∈ −≤1∈z/2Z 1x X∈ ∈ ∈∈ ≤ · · · ∈ · ····∈x· X≤z Z ∈ ∈ ∈ ∈si cons siv(conSconi)vfor(vSNon(S)ali)forlcforion=alvallexit1,il.i=.y=. 1,ofm,1.,t,..he..,.m,smu,∈m,n is∈caused by a small number (n + 1) of sets Shapley-Folkman Theorem: Let Si∗=DuSalit1 +2y Gapi +DecomcSm withpositionSi , q(µ) = If sminconLevvx(S1el)−sthenxet2 +!sxµ=(|xfs11(+x+)x2≤∈−+f1s)∈m+∈whereαc /·2·"· Optimal solu⊂tio#n sent x0 i =Shapley-F1x,1.≥. .0, xm2≥0 olkman Theorem:si LetSsiisfoi SrSatiS=ifolSfoer1astrat+at∗lmelasteast+nmSmm1nwithindn ice11Sindsiindi.iceices si.i,. n Shapley-F∈ ! olvkmv an Theorem:q·(∈·µ·) v=∈"Con∈LetmvinexS andx=x·S1·conca1·−+x2−v+−e−+µpar(−Sx−m1t +canwithx2⊂b−eS#1esi)timated, separately i =si 1, .con. ,vm(Si) for alxl ih=(λ1), .h.1.(,λm),Theh2(λsThe∗u0)Them−ofshu0s1mau(mλlargeof∗)ofaxalarge∗n·large·um·(−bnλerum∗n,umof1b)erbcon(er−ofvf˜λofex(con,x1con))s⊂ets=vexv#(exclisˇsets)salmfets(xis)osisalmalmt conosostvetconxconvevxex If s con−∈µiv=(S1i)f, then.µ. .≥, m1s = s1 + + sm wherexq1≥is ,cxlose2≥ d and concave =If∈ssi conSi fov(rSat) thenl∗east∗ sm= s n· +· · 1 ind+∗ siceswherei. ! " SloSpleop=e −=sλ−k+λcon#1k+−S1S∈vl∞lo(oSppleeo)pi=fforeµ−=<alλλ−kl1+λi1S=loSS1pll,oeo.−pp.=e1.e,=−=m−λ−,−kλλkvkNonS+vml1ko+pc1peNonon(v=Nonpv)(exitv−cX)oncλonkyvexitvofvexitk+tyhe1yofpsofu(tvhemt)heissuscmuaumissisedcaucbauysedsaedsmallbybyaasmallnsmallumbnerumnum(nberb+er1)(n(nof++1)se1)tsofofsesetsts i ∈If si conv(S) then s =·s·1· + Mi+−snµmComwhereifmonµ ≥ cP1 robl2em ∈si conThev(∈Sis)uform ofallailarge= 1, .n.um. , mber, ·=of· · convex psx(etsvk) +is alm"v"ost convex si Si fosri at conleastv(Smi) forn all1iind= ice1, .s. .i., m, −∞ if µ < 1 ˇ ∈ Nonconvexity of the sum is cau#Maxsed bCrossingy a small2Prnoblumebmer (nq+∗ =1)(ofcl )sep(0)ts p(0) = w∗ ∈si Si fo∈r at least−m −n 1 indices i. f˜(x) =f˜(f˜x((clˇ)x)=)f=(clxˇ()clˇ)f)(fx()x) q(Theµ)s=isummSiniofffo1|a(xrfx11largeat)|(x+)lfexast22n(fxum+2)(mµxb)xer1f)n1of(xcon)1 indvfex2W(icexes)aketss iDu.is alitalmyosq t conwvex ≤ ∈ x2≥0 − − ∗ ∗ 1 NonThecon∈vsexitum! yofofatlargehe sunmum−isbc"erau−ofsedconbyvxexaksmalls−etsDuαkisalitngumalmkybGaperos≤(tnDecomcon+ 1)vexofpositionsets The sum of a large n˜umberq(µofˇ) con= mvinexinf Ls|etsxc(1x|is,+λalmkx)2 +osµtx1con) vex (a)(a()b)(b) (a) (b)Nonc0onvexitify|µof| ≤the1 sum fis(xcau) =sed(clb)yf(axx)smallX number (n + 1) of sets = Nonconvexity of the sum is causedxCon2∈≥by0 vaexsmalland nconcaminiumˇ bmerviezˇ(eparˇn +wt 1)canofbsee tsestimated separately # −∞ if |µ| ≥ 1 xk+1 = PX (xk!− αqk∗ g=k)q(cl∗q∗=)p=(0)(cl(cl")p)(0)pp(0)(0) =p(0)pw(0)∗==ww∗ ∗ f˜(x) = (clˇ )f(x) q0is closeifd|µand| ≤ conca1 v≤e ≤≤ ˜ 1 ˇ 2 subject to (0, w) M, minmiminizmeize f(xf)(mx=)inx=i1mx+i1zex+2 xf2 (fx()xˇ=) =Dux˜ (clalit+)xfyDu=(xGapˇ)alitMiynDecomGapComv DecommonpositionProblpositionem ∈ %q∗ = (cl )pf(0)(x) =Dup(alit(0)cl#)fy−=(∞Gapxw)∗ iDecomf |µ| ≥p1osition q(µ) = infr p(u) + µ u ≤ subsujebcjtectot toug∈(x#g)(sx=u)bx=je1cx≤t1t0≤o 0g(x) = Conx1 ≤vConex0ConandvMaxexv∗exconcaandandCrossingconcavconcae parvtevPrecanparoblpartbetcanmecanestibebmeesatedestitimmseatedatedparatelyseseparatelyparately ! Slope =ˇ"g0 Slope = g1 Sxlope = g2 vk+1 p(v) Duality Gap Decomq∗ = (clposition)p(0) p(0) = wW∗ eak Duality q w x ∈xX∈ X= =(x1q(,xx1=2,x)x(∈2|clˇ)xX)|1px>(0)=≤q1 0is>}(c0xqqlosep}1(q(0),µisx)isd2c=)losecand|losewxind1fconcad>andandp0(}uconcav)concae+ µ∗v%euve ∗ Convex and conca∗ve parq t=can(clˇb)pe(0)estimpated(0)u∈∗#=rsepwarately ≤ Duality Gap(λ,Decomµ, 1)! p!osition ∗ Mi!n≤ComMiMin≤nmonComComPmonmonroblP∗emProblroblemem q is closed and concave f1(x) !f2(x) " minimize w ConDuvexalitfand∗Duyf=∗Gapalitconca∞= ,y∞DecomqGap,∗veq=∗par−Decom=pf∞t−osition∗can∞= p∞bosition,eMaxqes∗ti=mMaxCrossing−atedMax∞xCrossingsekCrossingparatelyProblPremProbloblemem Min Com1mon Problem q isConclosevexd andand(a)concaconca(b) vvee part canWbeakesWtiDuWemakealitatedakDuyDu(λalitseq∗,alitpµaratelyy, y1qw)∗q∗∗ ww∗ ∗ subject to (0, w) M, MaxConCrossingvex andPrconcaoblemve part can be estimated≤ sep≤arately≤ ∈ Minq Comis closeqmonisdcloseandProbldconcaandem vconcae ve 1 Weak Du2alit2 y q w m2inimize f(x) = x1 +minix2 mminiiminize mmwizieze ww MaxMiCrossingSn =ComSMi=n(xmonCom(Pr,xxobl),Pmonx|robl)e∗xm|S≤>xePm=robl0>∗ 0(exm, x) | x > 0 1 Max Crossing! ! Problem"! "subject to g"(x) = x1 ≤sub0 jectsubsubtojectject(0to, tow)(0(0, wM, w),) MM, , Weak DuMaxality Crossingq∗ w∗ Prminioblemmize w ∈ ∈∈ Weak Duality≤q∗ 2w∗ 2 q(µq)(µ=)Wm=eiaknminxDu+xalitqµ+(≤xµyµ)xq=2∗ miwn∗ x + µx x ∈ X = (x1, x2) | x1 > 0} x∈#x∈# minimsub≤ixz∈eject# wto (0, w) M, ! ! "mini" mi!ze w " ∈ ! −1−/(14/µ()4subµ)ifjectµif>µto−mini0>1/(00(m4, wµiz))e ifMw∗µ,=>∞0 , q∗ = −∞ = = = ∈ # −#∞−∞ ifsubµif#≤jectµ−sub0≤∞to0ject(0,towi)f(0µ,≤Mw),0 M, ∈ ∈ 2 minmiminizmeize f(xf)(mx=)in−=imx−izxe f(x) = −Sx = (x , x) | x > 0 subsujebcjtectot tog(xg)(sx=u)bx=je−cxt1−t/o21/≤2g(0≤x)0=
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