Interior point method for nonlinear nonconvex optimization Ladislav - PowerPoint PPT Presentation

Interior point method for nonlinear nonconvex optimization Ladislav Lukšan, Ctirad Matonoha, Jan Vlˇ cek Institute of Computer Science AS CR, Prague GAMM Workshop Applied and Numerical Linear Algebra September 11-12, 2008 Technische Universität Hamburg-Harburg, Germany

Outline 1. Introduction 2. Direction determination I. 3. Indefinitely preconditioned CGM 4. Linear dependence of gradients of active constraints 5. Numerical experiments 6. Direction determination II. 7. Linear dependence of gradients of active constraints L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 2

1. Introduction L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 3

General nonlinear programming problem Consider the general nonlinear programming problem (NP) x = arg min x ∈R n f ( x ) subject to c I ( x ) ≤ 0 , c E ( x ) = 0 , where c I ( x ) = [ c i ( x ) : i ∈ I ] T , I = { 1 , . . . , m I } c E ( x ) = [ c i ( x ) : i ∈ E ] T , E = { m I + 1 , . . . , m I + m E = m } . We assume that the functions f ( x ) : R n → R , c I ( x ) : R n → R m I , c E ( x ) : R n → R m E are twice continuously differentiable. L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 4

KKT conditions for (NP) The necessary KKT (Karush-Kuhn-Tucker) conditions for the solution of problem (NP) have the following form: g ( x, u ) = 0 , u T c I ( x ) ≤ 0 , u I ≥ 0 , I c I ( x ) = 0 , c E ( x ) = 0 , where g ( x, u ) = ∇ f ( x ) + A I ( x ) u I + A E ( x ) u E , and A I ( x ) = [ ∇ c i ( x ) : i ∈ I ] , A E ( x ) = [ ∇ c i ( x ) : i ∈ E ] . Here u I = [ u i ( x ) : i ∈ I ] T , u E = [ u i ( x ) : i ∈ E ] T are vectors of Lagrange multipliers. 5 L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method...

The idea of interior point methods We introduce of a slack vector s I = [ s i ( x ) : i ∈ I ] T and transform original problem (NP) to the sequence of problems with the logarithmic barrier function f ( x ) − µe T ln( S I ) e � � (IP) x = arg min , ( x,s I ) ∈R n + mI subject to c I ( x ) + s I = 0 , c E ( x ) = 0 , where µ > 0 is a barrier parameter, e is the vector with unit elements, and S I = diag( s i : i ∈ I ) . ● The logarithmic barrier term is used to ensure the inequality s I ≥ 0 implicitly. ● If µ = 0 , then the KKT conditions for (IP) coincide with the KKT conditions for (NP). Therefore µ → 0 is assumed. L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 6

KKT conditions for (IP) The necessary KKT conditions for the solution of problem (IP) have the following form (primal-dual formulation): g ( x, u ) = 0 , S I U I e − µe = 0 , (1) c I ( x ) + s I = 0 , c E ( x ) = 0 , where U I = diag( u i : i ∈ I ) . Inequalities s I > 0 and u I > 0 are required in all iterations. ● condition s I > 0 is necessary for the definition of the logarithmic barrier function, ● condition u I > 0 improves the properties of the linear system solved and is necessary for the construction of an efficient preconditioner. 7 L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method...

Newton’s method Linearizing the primal-dual equations, we get one step of the Newton method G 0 A I A E ∆ x g       0 U I S I 0 ∆ s I S I U I e − µe  = −  , (2)       A T I 0 0 ∆ u I c I + s I     I A T ∆ u E c E 0 0 0 E where g = g ( x, u ) and � � G = G ( x, u ) = ∇ 2 f ( x ) + u i ∇ 2 c i ( x ) + u i ∇ 2 c i ( x ) . i ∈ I i ∈ E ● The Hessian matrix G ( x, u ) is not usually given analytically, but automatic or numerical differentiation is used instead. ● We assume that the matrix of system (2) is nonsingular. L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 8

Description of algorithm The algorithm for an interior point method can be roughly described in the following form. x ∈ R n , s I ∈ R m I , u I ∈ R m I , u E ∈ R m E 1. Let vectors such that s I > 0 , u I > 0 be given. 2. Let a barrier parameter µ > 0 be given. 3. Determine direction vectors ∆ x, ∆ s I , ∆ u I , ∆ u E by solving a linear system equivalent to (2). 4. Choose a step-length 0 < α ≤ α . 5. Set x := x + α ∆ x, s I := s I ( α, ∆ s I ) , u I := u I ( α, ∆ u I ) , u E := u E + α ∆ u E , where s I ( α, ∆ s I ) > 0 and u I ( α, ∆ u I ) > 0 are functions of α depending on ∆ s I and ∆ u I , which are chosen by a suitable strategy. 6. Determine a new barrier parameter µ > 0 . L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 9

2. Direction determination I. L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 10

Active and inactive constraints KKT condition (1) implies that S I U I e ≈ µe and if µ → 0 , then either u i → 0 or s i → 0 holds for every index i ∈ I. Therefore, we can split the set of inequality constraints to an active and inactive subsets. Active: s i ≤ ε I u i , i ∈ I – denoted by ˆ ., i.e. ˆ c I ( x ) , ˆ s I , ˆ u I . Active constraints are those for which c i ( x ) , i ∈ I, are close to zero, c I ∈ R ˆ m I . where ˆ Inactive: s i > ε I u i , i ∈ I – denoted by ˇ ., i.e. ˇ c I ( x ) , ˇ s I , ˇ u I . Inactive constraints are those for which u i , i ∈ I, are close to zero, u I ∈ R ˇ m I . where ˇ Here ε I > 0 is a suitable parameter and ˆ m I + ˇ m I = m I . A general definition of the set of indices of active constraints: ¯ E ( x ) = E ∪ { i ∈ I : c i ( x ) = 0 } L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 11

Elimination of ∆ s I System (2) is nonsymmetric with the dimension n + m E + 2 m I . This system can be symmetrized and reduced by the elimination of the vector ∆ s I . One has ∆ s I = − U − 1 S I ( u I + ∆ u I ) + µU − 1 e I I so that       G A I A E g ∆ x − U − 1 A T  = − c I + µU − 1  . S I 0 ∆ u I e (3) I I     I A T ∆ u E c E 0 0 E Disadvantage: elements of matrix U − 1 S I can be unbounded, since u i → 0 I if the i -th inequality constraint is inactive at the solution point. L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 12

Elimination of inactive equations By elimination of inactive equations we obtain u I = ˇ ˇ c I + ˇ I ∆ x ) + µ ˇ S − 1 S − 1 A T ∆ˇ U I (ˇ I e I so that ˆ ˆ       ˆ g G A I A E ∆ x  = −  , ˆ − ˆ ˆ c I + µ ˆ U − 1 U − 1 A T ∆ˆ u I S I 0 ˆ e (4)     I I I ∆ u E A T c E 0 0 E where ˆ G + ˇ A I ˇ U I ˇ ˇ S − 1 A T G = I , (5) I g + ˇ A I ˇ ˇ c I + µ ˇ A I ˇ S − 1 S − 1 g ˆ = U I ˇ I e. (6) I L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 13

Boundedness of matrices Both matrices ˆ G and ˆ ˆ U − 1 S I are bounded (if G and A are bounded) and if I the strict complementarity conditions µ → 0 ( s i + u i ) > 0 , lim i ∈ I, hold (recall that s i > 0 and u i > 0 ), then one has ˆ ˆ U − 1 lim S I = 0 . I µ → 0 Similarly, the matrix ˇ ˇ S − 1 U I is bounded and if the strict complementarity I conditions hold, then ˇ ˇ S − 1 lim U I = 0 . I µ → 0 L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 14

Splitting of ∆ s I At the same time, we can split equality for ∆ s I into two equalities to obtain − ˆ ˆ u I ) + µ ˆ U − 1 U − 1 ∆ˆ s I = S I (ˆ u I + ∆ˆ e, I I c I + ˇ A T ∆ˇ s I = − (ˇ I ∆ x + ˇ s I ) after re-arrangements. Elimination of inactive constraints is quite a general approach: ● if ε I is large enough, we obtain original system (3); ● if ε I is close to zero, all constraints are inactive. A choice of a suitable ε I can improve effectiveness of the algorithm and decrease the number of operations in an iterative method. L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 15

3. Indefinitely preconditioned CGM L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 16

Indefinite system To simplify the notation, we rewrite system (4) containing only active constraints in the form � ˆ ˆ � � � � � d b G A K ¯ = ¯ d = = b, (7) ˆ ˆ ˆ − ˆ A T d b M where ˆ A = [ ˆ A I , A E ] and ˆ M = diag( ˆ M I , 0) . Here ˆ M I = ˆ ˆ U − 1 S I is a positive I definite diagonal matrix. We assume that matrix K is nonsingular, which implies that A E has a full column rank (gradients of active constraints are linearly independent). System (7) is symmetric and indefinite of order n + ˆ m = n + ˆ m I + m E . It can be solved ● either directly by using the sparse Bunch-Parlett decomposition ● or iteratively by using Krylov-subspace methods for symmetric indefinite systems. L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 17

The preconditioner We use a nonsingular preconditioner � � ˆ ˆ D A C = , ˆ − ˆ A T M where ˆ D is a positive definite diagonal matrix derived from the diagonal of G. We restrict to the situation when matrix ˆ ˆ G − ˆ D is non-singular (a usual situation and the worst case in some sense). One has � � I + ( ˆ G − ˆ D ) ˆ ( ˆ G − ˆ D ) ˆ P Q KC − 1 = , 0 I where D − 1 − ˆ D − 1 ˆ A T ˆ D − 1 ˆ M ) − 1 ˆ A T ˆ P = ˆ ˆ A ( ˆ A + ˆ D − 1 , D − 1 ˆ A T ˆ D − 1 ˆ Q = ˆ ˆ A ( ˆ A + ˆ M ) − 1 . L.Lukšan, C.Matonoha, J.Vlˇ cek: Interior point method... 18