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o#2$|r_dL%ɮ#`#2$sH%]m#2$^u3.XhM# 0e0e     A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`f3f3@S"8 g4MdMd0 ppp@ <4!d!d(z<4BdBd(z___PPT9zk{Xb10h'0? +  SIAM00O =^4Interior-Point Methods and Semidefinite Programming25(  IYin Zhang Rice University SIAM Annual Meeting Puerto Rico, July 14, 2000JJ'Outline0The problem What are interior-point methods? Complexity theory for convex optimization Narrowing the gap between theory and practice How practical are interior-point methods? This presentation is focused on a brief overview and a few selected topics will inevitably omit many important topics and works@nnPart 1. The problem Constrained optimization Conic programming They are  equivalent DDConstrained Optimization (CO) JConic Programming (CP)oCP has a linear objective function Constraint set Q = {affine space} {cone} Difficulties hidden in the cone$p1</0 Natural Conic ProgramsLP --- nonnegative orthant: SDP --- semidefinite matrix cone: SOCP --- second order (ice cream) cone: Direct sums of the above(?B1 Linear Objective Function 2 0Q Affine Space Cone$ k0Q Affine Space Cone$ + Convex or NonconvexxComplexity theory exists for convex programs f and Q are convex, or K is convex Local minima will all be global -&d&d &*1Part 2: What are Interior-Point Methods? (IPMs)?v2  ^ Main ideas Classifications: Primal and primal-dual methods Feasible and infeasible methods.AA0Main Ideas: Interior + NewtonCP: min{<c,x> : Ax = b, x in K } Keep iterates in the interior of K Apply Newton s method (How?)$A!$Newton s MethodSquare Nonlinear systems: F(x) = 0 (#equations = #variables) Unconstrained optimization: Sufficient when f is convex Equality constrained optimization: 1st order necessary (KKT) conditions form a square system F(x,y) = 0 Inequality constrained optimization: KKT system with inequalities How do we apply Newton s method?Z&ZZZ#ZEZ%Z@Z  #7 %  ]pPrimal IPMs (Barrier Methods)lConsider CP: min{<c,x> : Ax=b, x in K } Solutions are necessarily on the boundary Barrier function: Subproblems for t > 0 (equalities only): min {<c,x> +t F(x): Ax = b } x(t) Under suitable conditions, the so-called central path x(t) exists, and as t 0, x(t) x* Newton s method becomes applicable )*+$ *5$s!Log-Barriers have a long history lEarly works on Log-barrier methods: Frisch 1955 Fiacco & McCormick, 1968 Murray, Wright, 70s Convergence results exist (e.g. x(t) x* ) No computational complexity results Modern complexity theory for IPMs: Karmarkar on LP, 1984 Many authors on LP, QP, LCP, SDP General theory: Nesterov and Nemirovskii, 1993~$#f$Z$#fb$ > 2Primal and Dual Conic ProgramsPrimal and Dual CPs: (P): min { : Ax = b, x in K } (D): max { : A*y + s = c, s in K* } where A* is the adjoint of A , < Ax, y > = < x, A*y > and K* is the dual cone of K , Weak Duality holds:^M  3Primal-Dual IPMsThey solve the primal and dual together Most efficient in practice First proposed for LP (Kojima/Mizuno/Yoshise 1990) They require strong duality: < c, x* > = < b, y* > Strong duality holds under reasonable conditions for the usual cones (LP, SDP, & & )(1j(1Sh#ASemidefinite Programming (SDP).Optimization over matrix variables Applications: Systems and Control theory, statistics Structural (truss) optimization Combinatorial optimization & & & & .1ZgZ1g6SDP (continued):pOptimality conditions: X, S in K Primal feasibility: A(X)  b = 0 Dual feasibility: A*(y) + S  C = 0 Complementarity: XS = 0 Primal-dual methods for SDP: Keep X, S in K (positive definite) Perturb and apply Newton to equalities Keep an eye on the P-D central path (X(t),y(t),S(t))<"xL5  ;7An SDP Complication4The system is non-square A(X)  b = 0, A*(y) + S  C = 0, XS  tI = 0 Many remedies: Helmberger/Rendl/Vanderbei/Wolkowisz, Kojima/Shida/Hara, Monteiro, Nesterov/Todd, Alizahde/Haeberly/Overton,& & Polynomial complexity bounds established A unification scheme: (Monteiro, YZ, 1996)Z.ZZZ+Z     .    @    L ?IPMs: Feasible vs. Infeasible32CP: min{ : Ax = b, x in K } We already require iterates to stay in K. How about the affine space? Feasible IPMs: Require iterates to stay in the affine space Easier to analyze, stronger results Infeasible IPMs: Not require iterates to stay in the affine space Easier to implement, more practicallZZRZZUZ' RU @Feasible and Infeasible IPMs2 HCP: min{ : Ax = b, x in K } e.g., 3x+y+2z = 1, K = {(x,y,z) >= 0}X#& !)2Part 3: Complexity theory for convex programmingWTwo wings make IPMs fly: In theory, they work great In practice, they work even better.??5Theory for Primal IPMsTFor t 0, apply Newton s method to: min { <c,x> + t F(x) : Ax = b } F and K hold the keys Q1: What barriers are good for Newton? Q2: What cones permit good barriers?&!L #   "  | Self-concordant Barrier Function,Strictly convex function in interior of K : The Hessian F  varies slowly (good for Newton) The gradient F is bounded in a special norm (implying F varies slowly near the central path) They guarantee good behavior of Newton on the function <c,x> + t F(x) for varying t,1aU' 1)'7        ]}ExamplesNonnegative Orthant: Symmetric, positive semidefinite cone: Log-barriers are optimal (achieving smallest theta value possible)~9Complexity Results for Primal IPMs (N&N 1993, simplified)$:$ hAssume Worst-case iteration number for : short-step methods long-step methods Different strategies exist to force t 0 A gap exists between theory and practicel5-S5,$)&BElegant theory has limitationsSelf-concordant barriers are not computable for general cone Polynomial bounds on iteration number do not necessarily mean polynomial algorithms A few nice cones (LP, SDP, SOCP, & ) are exceptions 48General Theory for Primal-Dual IPMs (Nesterov & Todd 98)$9% %  zTheory applies to symmetric cones: Convex, self-dual (K = K*) , homogeneous Only 5 such basic symmetric cones exist LP, SDP, SOCP, & , are covered Requires strong duality: <c, x*> = <b, y*> Same polynomial bounds on #iterations hold Polynomial bounds exist for #operations A gap still exists between theory & practice#oU+U-6Part 4: Narrowing the Gap Between Theory and PracticeF Infeasible algorithms Asymptotic complexity (terminology used by Ye),..;1Complexity of Infeasible Primal-Dual algorithms:All early complexity results require feasible starting points (hard to get) Practical algorithms only require starting points in the cone (easy) Can polynomial complexity be proven for infeasible algorithms? Affirmative answers would narrow the gap between theory and practice&?E'C"The answers are indeed affirmativeFor LP: YZ 1992, also for SDP: YZ 1996 Numerous works since 1992 Polynomial bounds are weaker than feasible case There are many infeasible paths in the cone, e.g., ds3 S3 <Asymptotic ComplexityWhy primal-dual algorithms are more efficient than primal ones in practice? Why long-step algorithms are more efficient than short-step ones in practice? Traditional complexity theory does not provide answers An answer lies in asymptotic behavior (i.e., local convergence rates)}=IPMs Are Not Really NewtonianWNonlinear system is parameterized Full steps cannot be taken Jacobian is often singular at solutions Can the asymptotic convergence rate be higher than linear? Quadratic? Higher? Affirmative answers would explain why far less iterations taken by good algorithms than predicted by worst-case bounds A fast local rate accelerates convergence 2XfP0DAnswers are all affirmativenLP: Quadratic and higher rates attainable YZ/Tapia/Dennis 92, YZ/Tapia 93, YZ/Tapia/Potra 93, Ye/Guler/Tapia/YZ 93, Mehrotra 93, YZ/D.Zhang/96, Wright/YZ 96, & & Extended to SDP and beyond IPMs can be made asymptotically close to Newton method or composite Newton methods Idea: fully utilizing factorizationst+wn%+2D>U!%Part 5: Practical Performance of IPMs2Remarkably successful on  natural CPs IPMs in Linear programming: Now in every major commercial code Brought an end to the Simplex era SDP: enabling technology Are there efficient interior-point algorithms for general convex programs in practice? How about for nonconvex programs?BCECEy#"Nonlinear ProgrammingKKT conditions form a nonlinear system with non-negativity constraints Interior-point framework: Perturb and Apply Newton Keep iterates in the cone 4a4G34#KKT system and PerturbationOptimality (KKT) conditions:u Does it work?General convex programming: Yes, provided that derivatives are available and affordable Global convergence can be established under reasonable conditions (but not poly. complexity) (e.g. El-Bakry/Tsuchiya/Tapia/YZ, 1992, & & .) Nonconvex programming: Yes, locally speaking (local optima, local convergence) Continuing research (Session MS68 today)0%) &etRecent Books on IPMsNesterov and Nemirovskii,  Interior Point Methods in Convex Programming , SIAM 1993 Wright,  Primal-Dual Interior Point Methods , SIAM, 1997 Ye,  Interior Point Algorithms: Theory and Analysis , John Wiley, 1997 Roos/Terlaky/Vial,  Theory and Algorithms for Linear Optimization: An Interior Point Approach , John Wiley, 1999 Renegar,  Mathematical View of Interior Point Methods in Convex Programming , SIAM, 2000 Also in many new linear programming booksZZ g|>A A 15-line MATLAB code for LP min {: Ax = b, x >= 0 }\B$     St0=cputime; [m,n]=size(A); x=sqrt(n)*ones(n,1); y=zeros(m,1); z = x; p = symmmd(A*A'); bc = 1 + max(norm(b),norm(c)); for iter = 1:100 Rd=A'*y+z-c; Rp=A*x-b; Rc=x.*z; residual=norm([Rd;Rp;Rc])/bc; fprintf('iter %2i: residual = %9.2e',iter,residual); fprintf('\tobj=%14.6e\n',c'*x); if residual<5.e-8 break;end; gap=mean(Rc); Rc=Rc-min(.1,100*gap)*gap; d=min(5.e+15,x./z); B = A*sparse(1:n,1:n,d)*A'; R = cholinc(B(p,p),'inf'); t1 = x.*Rd - Rc; t2 = -(Rp + A*(t1./z)); dy = zeros(m,1); dy(p)=R\(R'\t2(p)); dx=(x.*(A'*dy)+t1)./z; dz=-(z.*dx+Rc)./x; tau = max(.9995, 1-gap); ap = -1/min(min(dx./x),-1); ad = -1/min(min(dz./z),-1); ap = tau*ap; ad = tau*ad; x = x + ap*dx; z = z + ad*dz; y = y + ad*dy; End fprintf('Done!\t[m n] = [%g %g]\tCPU = %g\n',m,n,cputime-t0); Txe"""""|""=$"" (!   ?P     0   /  P ` 33PP` 3333` ___MMM` 13` 333fpKNāvI` j@v۩ῑ΂H` Q_{>?" dd@$?n<d@`7 `2@`7``2 n?" dd@   @@``PR    @ ` ` p>>  K0    P (  P P <$ D "  " P T$d D "  " P <$U_ D "  " P TT$d>& D "  " P NT$P D "  " P <$p D "  " P C x$?d ?bUv D "  "  P 64$ U A T Click to edit Master title style! !"  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F@8 '$D0H 0 0޽h ? 3333@   dn(  d d s *A PU<$D 0  x " d c $A ??@@ 8 $D 0 d c $A E??P @B8 E$D 0N d 0A  0 ,$ 0 (W.L.O.G, we can assume f(x)=, i.e.,R) n<    "B d s *޽h ? www3ff    VNh(  h h s *TA PU<$D 0   "  h c $A" ?? 8 $D 0` h c $A ??& Z P B h s *޽h ? www3ff   ]U (  r  S tA PU     c $A ??0 8 $D 0`   c $A s?? F h s?  <A ?@l,$D 0 +Hence, CO = CP (theoretically convenient)J,0 n<Z "H  0޽h ? 3333   `D(  ` ` s *4A PU<$D 0   " ` s *A P<$ 0  " ` c $A ??I8 $D 0B ` s *޽h ? www3ffl   (    s *A TpP<$D 0    s *tA T0@<$ 0  H  0޽h ? 3333  08(  8 8 s *A TpP<$ 0   8 s *$ħ T```<$ 0  "P@0 pH 8 0޽h ? 3333d   @(  r  S ħ PU   r  S ħ P   `  c $A ??@ H  0޽h ? 3333   d\P(  r  S Ƨ PP   r  S dƧ P  `  c $A ??0  `  c $A ??P  H  0޽h ? 3333d   `(  r  S Dȧ PU   r  S ȧ P   `  c $A$ C??`  CH  0޽h ? 3333   phT(  Tx T c $ɧ PU   x T c $$ʧ P   ` T c $A ?? :  ` T c $A% y?? 0  yH T 0޽h ? 3333   X0(  Xx X c $ʧ PU   x X c $D˧ P   H X 0޽h ? 3333    ,(  x  c $[ PP  x x  c $T\ P0   x   c $A! ??08 $D 0`  c $A ??  H  0޽h ? 3333   ( p(  pr p S ̧ PP   r p S ḑ P    p c $A( }??@ 8 8 }$D 0H p 0޽h ? 3333p   t(  tx t c $DΧ PU   x t c $Χ P   x ` t c $A ??  H t 0޽h ? 3333l   (    s *x PU<$D 0  "   s *4x P <$ 0 " H  0޽h ? 3333    :(    c $x PU<$D 0  "   c $Tx P` <$D 0 "   c NA*28 \D:\Yin\docs\pic2.bmpP`0,$D  0H  0޽h ? 3333|   (    s *tx TpP<$D 0 "   s *x Tp`  <$ 0 " "B  s *޽h ? www3ff   91d(  dx d c $4x PU  " x d c $x P   "  d <x ? ,$D 0 General theory by Nesterov & Nemirovskii: A1: Self-concordant barrier functions A2: Essentially all convex cones (that contain a non-empty interior but no lines)*0 n<ZI0 n<Z1 " 7*333 0, |"H d 0޽h ? 3333   d\ (  r  S x PU  " r  S 4x P  " `  c $A ?? P  `  c $A ??  H  0޽h ? 3333   d\0(  r  S x PU  " r  S x P  " `  c $A ??P `  c $A# ?? ,  H  0޽h ? 3333J   @r(  r  S 4x PU  "   S x P  " "P@0 p`  c $A ??0 `  c $A ??P  `  c $A 3??0 * 3H  0޽h ? 3333   `$(  r  S T" PU  " r  S " P  " H  0޽h ? 3333   p\0(  \x \ c $" PU  " x \ c $" P  " H \ 0޽h ? 3333f  $(  $ $ s *" TpP<$D 0 "  $ s *T" T `  <$ 0 " B $ s *޽h ? www3ff   $(  r  S " PU   r  S " P   H  0޽h ? 3333L   t(  r  S " PU   r  S " P     c $A ~?? 8 ~$D 0  c $A) ?? ! 8 $D 0  c $A' ?? Pp 8 $D 0  c $A& ?? @:8 $D 0H  0޽h ? 3333   $(  r  S " PU   r  S " P   H  0޽h ? 3333   $(  r  S " PU   r  S dn P   H  0޽h ? 3333p   (  x  c $p PU   x  c $q P   `  c $A* ??p p@ H  0޽h ? 3333   $(  r  S q PU   r  S $r P   H  0޽h ? 3333p   (  x  c $s PU   x  c $t P`  `  c $A ??m H  0޽h ? 3333.    V(  r  S t PU   r  S $u P00  `  c $A ??  .  <u ? @J\  Perturb KKT, then apply Newton Hopefully, (x(t),y(t),z(t)) (x*,y*,z*) (There is a close connection to log-barrier)O0 n<Z-0 n<Z/ -.M"   c $A ??@@ 8 $D 0H  0޽h ? 3333   0$(  r  S Dv PU   r  S v P   H  0޽h ? 3333   @$(  r  S w PU   r  S $x P   H  0޽h ? 3333   P$(  r  S x PU   r  S Dy P   H  0޽h ? 3333/ 0 4V(  4X 4 C <   A 4 S 4A< @  A XDNonnegative orthant cones are actually direct sums of 1-d SDP cones.H 4 0޽h ? ̙33 0 p`W(  `X ` C <    ` S dɧ< @   YELog-barriers are most popular and proven to be optimal in N&N theory.H ` 0޽h ? ̙335 0 h%(  hX h C <   " h S tx< @  " 'Mild qualificationsH h 0޽h ? ̙33 0 Pl`(  lX l C <   " l S "< @  " bNTheta is the complexity number of the barrier. Gap between theory and practiceH l 0޽h ? ̙334 0 @(  X  C <   "  S "< @  " B.There exist only 5 irreducible symmetric conesH  0޽h ? ̙33# 0  a(  X  C <     S u< @   c5Fiacco and McCormick did not apply Newton to the PKKT/H  0޽h ? ̙33  0 +(  X  C <     S A< @   -Q is embedded in R^{n+1} H  0޽h ? ̙33r@ 0 2*(  ^ S    "$ c $x @  " Note the paths, so-called central path. They do exists globally for convex cones and at least locally for nonconvex ones. We will introduce a definition shortly. 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