On Superlinear Convergence of Infeasible InteriorPoint Algorithms for Linearly Constrained Convex Programs *


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1 Computational Optimization and Applications, 8, (1997) c 1997 Kluwer Academic Publishers. Manufactured in The Netherlands. On Superlinear Convergence of Infeasible InteriorPoint Algorithms for Linearly Constrained Convex Programs * RENATO D.C. MONTEIRO FANGJUN ZHOU School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA Received July 18, 1995; Revised April 18, 1996; Accepted July 12, 1996 Abstract. This note derives bounds on the length of the primaldual affine scaling directions associated with a linearly constrained convex program satisfying the following conditions: 1) the problem has a solution satisfying strict complementarity, 2) the Hessian of the objective function satisfies a certain invariance property. We illustrate the usefulness of these bounds by establishing the superlinear convergence of the algorithm presented in Wright and Ralph [22] for solving the optimality conditions associated with a linearly constrained convex program satisfying the above conditions. Keywords: InfeasibleInteriorPoint algorithm, affine scaling, convex program, superlinear convergence 1. Introduction During the past few years, we have seen the appearance of many papers dealing with primaldual (feasible and infeasible) interior point algorithms for linear programs (LP), convex quadratic programs (QP), monotone linear complementarity problems (LCP) and monotone nonlinear complementarity problems (NCP) that are superlinearly or quadratically convergent. For LP and QP, these works include [1, 4, 5, 19, 23, 25, 27, 28, 29]. For LCP, we mention the papers [10, 11, 12, 13, 21, 24] and for NCP, we cite [3, 14, 15, 22]. In this paper we are interested in the superlinear convergence analysis of infeasible interior point algorithms for solving the linearly constrained convex program minimize x f(x) subject to Ax = b, x 0, (1) where x IR n,a IR m n,b IR m,f:ir n IRis a sufficiently smooth convex function, the feasible set {x Ax = b, x 0} is nonempty and m<n. A key result used in the superlinear convergence analysis of several feasible and infeasible interior point algorithms for convex QP and monotone LCP problems is the fact that the length of the primaldual affine scaling direction at a given primaldual infeasible interior point (x, s, y) IR 2n ++ IR m satisfying some centrality condition is bounded above by * The work was based on research supported by the Office of Naval Research under grants N and N
2 246 MONTEIRO AND ZHOU Cx T s, for some constant C > 0, whenever the problem has a solution satisfying strict complementarity. The goal of this paper is to show that the primaldual affine scaling directions associated with problem (1) also satisfies a similar bound whenever the Hessian 2 f( ) of the objective function f( ) satisfies a certain invariance property and the problem has a solution satisfying strict complementarity. The invariance property is satisfied by all functions of the form f(x) =u(ex)+c T x, where E IR l n, c IR n and u : IR l IRis a twice continuously differentiable function such that 2 u(y) > 0, for all y IR l. We illustrate the usefulness of these bounds by establishing the superlinear convergence of the algorithm presented in Wright and Ralph [22] for solving the (mixed) NCP (see relations (2)(6) below) determined by the optimality conditions associated with a convex program (1) satisfying the following two conditions: the Hessian 2 f( ) satisfies the invariance property cited above and (1) has a solution satisfying strict complementarity. We should mention that the bounds derived in this paper can also be used to establish the superlinear convergence of other algorithms for solving (2) (6). For example, Monteiro and Wright [8] develops a primaldual feasibleinteriorpoint algorithm for solving (2) (6) which can be shown to converge superlinearly with the help of these bounds. Since these bounds hold for both feasible and infeasible points, we chose an infeasibleinteriorpoint algorithm such as the one developed by Wright and Ralph as the focus of our presentation. An interesting question, which we do not attempt to answer in this paper, is whether the bounds derived here can also be used to establish the superlinear convergence of the infeasibleinteriorpoint method introduced by Kojima, Megiddo and Noma [3] without assuming the existence of a unique nondegenerate solution. The following notation is used throughout the paper. IR p, IR p + and IR p ++ denote the p dimensional Euclidean space, the nonnegative orthant of IR p and the positive orthant of IR p, respectively. The set of all p q matrices with real entries is denoted by IR p q. The diagonal matrix corresponding to a vector u is denoted by diag (u). The ith component of a vector u IR p is denoted by u i and, for an index set α {1,...,p}, the subvector [u i ] i α is denoted by u α.ifα {1,...,p}, β {1,...,q}and Q IR p q, we let Q αβ denote the submatrix [Q ij ] i α,j β ;ifβ={1,...,q}we denote Q αβ simply by Q α and if α = {1,...,p}we denote Q αβ by Q β or Q β. For a vector u, the Euclidean norm, the 1norm and the norm are denoted by, 1 and, respectively. Given a matrix Q IR p q, we let Range (Q) {Qv v IR q } and Null (Q) {v IR q Qv =0}.We say that (B,N) is a partition of {1,...,p}if B N = {1,...,p}and B N =. The superscript T denotes transpose. For u, v IR n, we let [u, v] {tu +(1 t)v:t [0, 1]} denote the line segment whose endpoints are u and v. 2. Description of the problem and the main results In this section we introduce the notation, terminology and assumptions to be used throughout the paper. We also state the main result of this paper on the existence of certain bounds on the length of the primaldual affine scaling directions associated with (1) when this problem has a solution satisfying strict complementarity and the Hessian 2 f( ) of the objective function of (1) satisfies a certain invariance condition. Finally, we discuss the implication of
3 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 247 these bounds to the superlinear convergence analysis of infeasible interior point algorithms for solving (1). It is wellknown that x IR n is an optimal solution of (1) if and only if there exists (y, s) IR m IR n satisfying the following firstorder optimality conditions for (1): A T y + s = f(x), Ax = b, x 0, s 0, x T s = 0. (2) (3) (4) (5) (6) We start by stating the assumptions that will be used throughout our presentation. Assumption 1 rank(a) =m; Assumption 2 the function f is convex and twice continuously differentiable; Assumption 3 there exists a partition (B,N) of {1,...,n}and a solution (x,s,y )of (2) (6) such that x B > 0 and s N > 0; Assumption 4 the subspace A(x) Null ( 2 f(x))) Null (A) is constant for every x IR n +. Assumptions 1 and 2 are quite standard. It is wellknown that Assumption 3 plays an important role in proving that a large class of primaldual interior point algorithms converges superlinearly. Monteiro and Wright [9] shows that this assumption is indeed necessary for superlinear convergence of methods that behave like Newton s method near the solution, including the one discussed in this note. We next discuss Assumption 4, which looks unusual at first sight. First note that Assumption 4 clearly holds when f is a convex quadratic function. More generally, it is easily seen that any function of the form f(x) =u(bx)+c T x, where B IR l n, c IR n and u : IR l IR is a twice continuously differentiable function such that 2 u(y) > 0, for all y IR l, satisfies Assumption 4. Conversely, under Assumption 2, it follows as a consequence of Lemma 5.1 of [7] that if the stronger condition that Null( 2 f(x)) be constant on IR n holds then f has the above form. In view of Assumption 4, from now on we denote the constant subspace A(x), for x IR n +, simply by A. Any vector d IR n can be written as d = d A + d, d A A, d A, (7) where A denotes the orthogonal complement of A. The primaldual affine scaling search direction ( x, s, y) at a given infeasible interior point (x, s, y) IR 2n ++ IR m is computed by applying one step of Newton s method to the
4 248 MONTEIRO AND ZHOU nonlinear system defined by (2), (3) and XSe =0, where X = diag (x), S diag (s) and e =(1,...,1) IR n. Hence S x + X s = SXe, (8) A x = (Ax b), (9) 2 f(x) x + A T y + s = ( f(x)+a T y+s). (10) Similar to the case in which f( ) is a convex quadratic function, bounds on the affine scaling directions ( x, s, y) in terms of the duality gap x T s (or a certain power of it with an exponent close to 1) is usually obtained for points in a neighborhood of the central path. We next define a neighborhood of the central path which generalizes two other neighborhoods that have been considered in the literature (see for example [2, 10, 11, 18]). Given parameters δ 1,δ 2 0,η 1 >0and η 2 0, let N (δ 1,η 1,δ 2,η 2 ) where ( ) r P r = r D Ñ (δ 1,η 1,ρ) (x, s, y) IR 2n ++ IR m : η 2 (x T s) 1 δ2 r, x i s i η 1 (x T s) 1+δ1, i =1,...,n, (11) ( ) Ax b s f(x)+a T. (12) y Observe that when δ 2 = δ 1 /(1 + δ 1 ), the above neighborhood reduces to the neighborhood ρ(x T s) r 1+δ1, (x, s, y) IR 2n ++ IR m : x i s i η 1 (x T s) 1+δ1, i =1,...,n, (13) where ρ η 1+δ1 2. The neighborhood (13) is used in the algorithm presented in [11]; it is a generalization of the feasible neighborhood obtained by setting ρ =0in (13), which was independently introduced in [10, 18]. On the other hand, if δ 1 = δ 2 =0, the neighborhood (11) reduces to a neighborhood which was introduced by Kojima, Megiddo and Mizuno [2] and subsequently used in several papers dealing with infeasibleinteriorpoint algorithms (see for example [5, 16, 19, 20, 21, 22, 26, 30]). Wright and Ralph [22] have discussed an algorithm for solving a monotone NCP and analyzed its superlinear convergence properties. A slight modification of their algorithm can be used to solve the (mixed) NCP (2)(6) as well, and hence, the convex program (1). It is not our intention to restate the algorithm of [22] here since the description of the method, besides being lengthy, is not important for our presentation; only the necessary aspects of the algorithm will be discussed. For the purpose of summarizing the properties of the algorithm in [22], we let {(x k,s k,y k )} denote the sequence of iterates generated by this algorithm for solving (2) (6) and let {r k }
5 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 249 denote the corresponding sequence of residual vectors obtained by (12) with (x, s, y) = (x k,s k,y k ). First, we observe that the sequence {r k } satisfies the property that r k [0,r 0 ] for all k 0. Second, for some η 1,η 2 >0, there holds: {(x k,s k,y k )} N(0,η 1,0,η 2 ). (14) Third, it is shown in [22] that, for every k sufficiently large, the search direction at the kth iteration is equal to the affine scaling direction computed by (8)(10) with (x, s, y) = (x k,s k,y k ); the iterations for which this property holds are called fast steps in [22]; on the other hand, the search direction used in a nonfast iteration, called a safe step in [22], is a linear combination of the affine scaling direction and the centering direction. The main result obtained in [22] can be summarized as follows: under Condition A stated below, every accumulation point of the sequence {(x k,s k,y k )}is a solution of (2) (6) and the sequences { r k } and {x kt s k } converge to zero Rsuperlinearly and Qsuperlinearly, respectively. Condition A: The algorithm generates a bounded sequence {(x k,s k,y k )}and there exists a constant C>0such that the affine scaling direction ( x k, s k, y k ) calculated via (8)(10) by setting (x, s, y) =(x k,s k,y k )satisfies ( x k, s k ) Cx kt s k, (15) for all k sufficiently large. The main goal of this paper is to show that relation (15) of Condition A holds when (1) satisfies Assumptions 1, 2, 3 and 4 and the sequence {x k } is bounded. Specifically, we show in the next section that the following result holds. Theorem 1 Suppose that Assumptions 1, 2, 3 and 4 hold and let X be a bounded set. Let a point (x 0,s 0,y 0 ) IR 2n ++ IR m and parameters δ 1,δ 2,η 2 0, and η 1 > 0 be given. Then, there exists a constant C 0 with the following property: for any (x, s, y) N (δ 1,η 1,δ 2,η 2 )and t [0, 1/2] satisfying Ax b = t(ax 0 b), (16) f(x)+a T y+s = t( f(x 0 )+A T y 0 +s 0 ), (17) and the conditions x T s min(1,x 0T s 0 ) and x X, the corresponding solution ( x, s, y) of (8)(10) satisfies max{ x, s } C(x T s) 1 δ, (18) where δ δ 1 +2δ 2. The proof of Theorem 1 follows as an immediate consequence of the more general result given in Theorem 2 (see the paragraph after the proof of Theorem 2). Using Theorem
6 250 MONTEIRO AND ZHOU 1, it is now easy to see that Assumptions 1, 2, 3 and 4 imply relation (15) of Condition (A) whenever {x k } is bounded. Indeed, since r k [0,r 0 ], it follows that conditions (16) and (17) are automatically satisfied by every iterate (x k,s k,y k ). Moreover, for every k sufficiently large, the iterate (x k,s k,y k )satisfies the other conditions of Theorem 1, due to the observations preceding Condition A. Hence, it follows from (18) that (15) holds. (Here, δ =0due to (14) and the fact that δ δ 1 +2δ 2.) Before ending this section, we observe that the requirement that the sequence (x k,s k,y k ) be bounded in Condition A is automatically satisfied in certain situations. We have the following result whose proof is given in the appendix; this result in the context of convex QP and monotone LCP is wellknown (see for example Lemma 2.1 of [9]). Proposition 1 Suppose that Assumptions 1 and 2 hold. For some constant t (0, 1), let {t k } k=1 [0, t] and {(x k,s k,y k )} k=0 IR2n ++ IR m be two sequences satisfying ( ) ( ) r k Ax k b Ax f(x k )+A T y k +s k = t 0 b k f(x 0 )+A T y 0 +s 0, k 1. Then the sequence {(x k,s k,y k )}is bounded whenever either one of the conditions below holds: (a) the sequence {x kt s k } is bounded and there exists a point ( x, s, ȳ) IR 2n ++ IR m satisfying relations (2) and (3); (b) there exists a constant ρ > 0 such that r k x kt s ρ r0, k 1. (19) k x 0T s0 3. Proof of the main result This section is devoted to the proof of Theorem 1. The main result of this section is Theorem 2, which is easily seen to imply Theorem 1. The following inequality is exploited in a number of proofs that follow. Lemma 1 Suppose that Assumption 2 holds. Let (x 0,s 0,y 0 )and (x, s, y) be points such that (16) and (17) are satisfied for some t [0, 1], and let ( x, s, ȳ) be a point such that A x b = 0, (20) f( x)+a T ȳ+ s = 0. (21) Then 0 t 2 x 0T s 0 +(1 t) 2 x T s+x T s+t(1 t) (x 0T s + x T s 0) t (x 0T s+x T s 0) (1 t) ( x T s + x T s ) + t(1 t) ( f(x 0 ) f( x) ) T (x 0 x). (22)
7 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 251 Proof. By (16), (17), (20) and (21), we have A(x tx 0 (1 t) x)=0, (s ts 0 (1 t) s)+( f(x)+t f(x 0 )+(1 t) f( x)) + A T (y ty 0 (1 t)ȳ)=0. Multiplying the second relation by [x tx 0 (1 t) x] T on the left and using the first relation, the fact that t [0, 1] and [ f(x 1 ) f(x 2 )] T (x 1 x 2 ) 0 for all x 1,x 2 IR n, due to Assumption 2, we obtain [ s ts 0 (1 t) s ] T [ x tx 0 (1 t) x ] = [ f(x) t f(x 0 ) (1 t) f( x) ] T [ x tx 0 (1 t) x ] = [ t ( f(x) f(x 0 ) ) +(1 t)( f(x) f( x)) ] T [t(x x 0 )+(1 t)(x x) ] = t 2 ( f(x) f(x 0 ) ) T (x x 0 )+(1 t) 2 ( f(x) f( x)) T (x x) [ ( f(x) + t(1 t) f(x 0 ) ) ] T (x x)+( f(x) f( x)) T (x x 0 ) t(1 t) [ ( f(x) f(x 0 )) T ( (x x 0 )+(x 0 x) ) +( f(x) f( x)) T ( (x x)+( x x 0 ) )] = t(1 t) ( f(x) f(x 0 ) ) T (x x 0 ) +t(1 t)( f(x) f( x)) T (x x) + t(1 t) ( f(x 0 ) f( x) ) T ( x x 0 ) t(1 t) ( f(x 0 ) f( x) ) T ( x x 0 ) Expanding this inequality, we then obtain (22). The next result is an immediate consequence of Lemma 1. Throughout our presentation, we use the following convention: the constants C i s are global ones while the constants L i s have meaning only locally within the proof of a result. Lemma 2 Suppose that Assumption 2 holds. Let a point w 0 (x 0,s 0,y 0 ) IR 2n ++ IR m be given. Then there exists a constant C 0 0 satisfying the following property: for any (x, s, y) IR 2n ++ IR m and t [0, 1] such that (16) and (17) are satisfied, there holds max{t x,t s } C 0 (x T s+t). (23) Proof. Let ( x, s, ȳ) be a solution of (2) (6). Using inequality (22) and the fact that t [0, 1], (x 0,s 0 )>0, (x, s) 0, ( x, s) 0, x T s =0and ( f(x 0 ) f( x)) T (x 0 x) 0, we obtain x 0T (ts)+s 0T (tx) tx 0T s 0 + x T s + t(x 0T s + s 0T x) +t ( f(x 0 ) f( x) ) T (x 0 x) = L 1 (x T s + t),
8 252 MONTEIRO AND ZHOU where L 1 1+x 0T s 0 +x 0T s+s 0T x+( f(x 0 ) f( x)) T (x 0 x). Relation (23) follows by letting C 0 = L 1 min i=1,...,n {min{x 0 i,s0 i }}. The next result gives a preliminary bound on the primaldual (scaled) affine scaling directions. Lemma 3 Suppose that Assumption 2 holds and let u 0,r 0 IR n,η 1 >0and δ 1 0 be given. Then, for every vector (x, s, y) IR 2n ++ IR m and scalar t 0 such that ( ) ( ) Ax b Au 0 s f(x)+a T = t y r 0, (24) min x is i η 1 (x T s) 1+δ1, (25) i=1,...,n the corresponding direction ( x, s, y) determined by (8)(10) satisfies ( x + tu 0 ) T ( s + tv(x))=( x+tu 0 ) T 2 f(x)( x + tu 0 ) 0 (26) and max ( D 1 x, D s ) (x T s) 1/2 + 2t η 1/2 1 (x T s) (1+δ1)/2 [ Su 0 + Xv(x) ], (27) where X diag (x), S diag (s), D X 1/2 S 1/2 and v(x) 2 f(x)u 0 +r 0. Proof. It follows from (9), (10) and (24) that A( x + tu 0 ) = 0, (28) ( s + t 2 f(x)u 0 + tr 0 )+A T y = 2 f(x)( x + tu 0 ). (29) Multiplying (29) on the left by ( x + tu 0 ) T and using (28) and Assumption 2, we obtain (26). To show (27), we first show that max ( D 1 x, D s ) (x T s) 1/2 +2t [ D 1 u 0 + Dv(x) ]. (30) Indeed, it follows from (26) that x T s tu 0T s tv(x) T x t 2 u 0T v(x). (31) Multiplying (8) on the left by (XS) 1/2 and squaring both sides, we obtain D 1 x 2 + D s 2 +2 x T s=x T s,
9 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 253 which, in view of (31), implies D 1 x tdv(x) 2 + D s td 1 u 0 2 t 2 Dv(x)+D 1 u 0 2 x T s. Hence, using the inequality (β 2 + γ 2 ) 1/2 β + γ and the triangle inequality, we obtain D 1 x [ x T s+t 2 Dv(x)+D 1 u 0 2] 1/2 +t Dv(x) (x T s) 1/2 +2t Dv(x) + t D 1 u 0 (x T s) 1/2 +2t [ Dv(x) + D 1 u 0 ]. The same bound for D s can be derived similarly, and hence (30) follows. The proof of (27) is now immediate. Indeed, using (25), we obtain D 1 u 0 2 = Similarly, we have n i=1 s i (u 0 i )2 x i = n i=1 (s i u 0 i )2 Su0 2 x i s i η 1 (x T. (32) 1+δ1 s) Dv(x) 2 Xv(x) 2 η 1 (x T. (33) 1+δ1 s) Relation (27) follows by substituting (32) and (33) into (30). The following result yields bounds on the nonbasic components of x and s. Lemma 4 Suppose that Assumptions 2 and 3 hold. Let a point (x 0,s 0,y 0 ) IR 2n ++ IR m and parameters δ 2,η 2 0be given. Then, there exists a constant C 1 0 with the following property: for any vector (x, s, y) IR 2n ++ IR m and t [0, 1 2 ] satisfying (16) and (17) and the conditions there hold x T s x 0T s 0 and η 2 (x T s) 1 δ2 r, (34) x i C 1 (x T s) 1 δ2, i N, (35) s i C 1 (x T s) 1 δ2, i B. (36) Proof. Let r 0 ( Ax 0 b s 0 f(x 0 )+A T y 0 ). (37) Here, we only consider the infeasible case: r 0 0. Let (x,s,y ) denote the point as in Assumption 3. Setting ( x, s) =(x,s ) in inequality (22) and using the fact that t [0, 1/2], (x, s) > 0, (x 0,s 0 )>0,(x,s ) 0and x T s =0, we obtain x T s + x T s t 1 t x0t s t xt s + t (x 0T s + x T s 0) + t ( f(x 0 ) f(x ) ) T (x 0 x ). (38)
10 254 MONTEIRO AND ZHOU By (16), (17) and (34), we have t = r r 0 η 2(x T s) 1 δ2 r 0. (39) Using (34), (38), (39) and the fact that 1 t 1/2, we obtain x T s + x T s 2x T s + (xt s) 1 δ2 r 0 η 2 (2x 0T s 0 + x 0T s + x T s 0 +( f(x 0 ) f(x )) T (x 0 x )) L 1 (x T s) 1 δ2, where L 1 2 ( x 0T s 0) δ r 0 η 2(2x 0T s 0 + x 0T s + x T s 0 +( f(x 0 ) f(x )) T (x 0 x )). This last relation immediately implies (35) and (36) if we define { C 1 L 1 max max i N 1 s i, max i B 1 x i }. We now use this lemma to bound the nonbasic components of ( x, s). Lemma 5 Suppose that Assumptions 2 and 3 hold. Let a point (x 0,s 0,y 0 ) IR 2n ++ IR m and parameters δ 1,δ 2,η 2 0and η 1 > 0 be given. Then, there exist two constants C 2,C 3 0 with the following property: for any (x, s, y) N(δ 1,η 1,δ 2,η 2 )and t [0, 1/2] satisfying (16) and (17) and the condition there hold x T s min(1,x 0T s 0 ), (40) x N ( C 2 +C 3 2 f(x) ) (x T s) 1 δ, (41) s B ( C 2 +C 3 2 f(x) ) (x T s) 1 δ, (42) where δ δ 1 +2δ 2. Proof. We only give the proof for the case in which r 0 0, where r 0 is given by (37). We first show that there exist two constants L 0,L 1 0such that max{ D 1 x, D s } (L 0 + L 1 2 f(x) )(x T s) (1 δ)/2. (43)
11 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 255 The assumptions of the lemma imply that (23) and (39) hold. These two relations together with (40) then imply max{t x,t s } C 0 (x T s+t) [ C 0 1+ η ] 2 r 0 (x T s) 1 δ2 = L 2 (x T s) 1 δ2, (44) where L 2 C 0 [1 + η 2 / r 0 ]. Recalling the definition of v(x) given in Lemma 3, we then have v(x) = 2 f(x)u 0 + r 0 2 f(x) u 0 + r 0. Since the assumptions of Lemma 3 are satisfied, inequality (27) together with (44) and (40) then yield max{ D 1 x, D s } (x T s) 1/2 2t [ + Su 0 + Xv(x) ] η 1/2 1 (x T s) (1+δ1)/2 (x T s) 1/2 2 ( + u 0 + v(x) ) max( tx η 1/2, ts ) 1 (x T s) (1+δ1)/2 (x T s) 1/2 2L 2 [( f(x) ) u 0 + r 0 ] (x T s) 1 δ2 η 1/2 1 (x T s) { (1+δ1)/2 1+ 2L 2 [( 1+ 2 f(x) ) u 0 + r 0 ]} (x T s) (1 δ1 2δ2)/2, η 1/2 1 where X diag (x) and S diag (s). This inequality clearly implies (43) upon letting L 0 1+2L 2 η 1/2 1 ( u 0 + r 0 )and L 1 2L 2 η 1/2 1 u 0 and noting that δ δ 1 +2δ 2. Considering i N and using the definitions D X 1/2 S 1/2 and δ δ 1 +2δ 2, the formulae (43) and (35) and the inclusion (x, s, y) N(δ 1,η 1,δ 2,η 2 ), we conclude that ( ) 1/2 xi ( x i L0 +L 1 2 f(x) ) (x T s) (1 δ)/2 = s i x i (x i s i ) 1/2 ( L0 + L 1 2 f(x) ) (x T s) (1 δ)/2 ( L 0 + L 1 2 f(x) ) C 1 (x T s) 1 δ2 η 1/2 1 (x T s) (1+δ1)/2 (xt s) (1 δ)/2 = ( C 2 + C 3 2 f(x) ) (x T s) 1 δ, (45) where C 2 nl 0 C 1 η 1/2 1 and C 3 nl 1 C 1 η 1/2 1. Hence, we have proved (41). The proof of (42) is identical. Bounding the remaining components of the search directions, namely x B and s N,is more difficult; we first need to establish five preliminary lemmas. The proof of the first lemma can be found in Monteiro and Wright [10]. It unifies Theorem 2.5 and Lemma A.1 of Monteiro, Tsuchiya and Wang [6], which in turn are based on Theorem 2 of Tseng and Luo [17].
12 256 MONTEIRO AND ZHOU Lemma 6 Let f IR q and H IR p q be given. Then there exists a nonnegative constant M = M(f,H) with the property that for any diagonal matrix D>0and any vector h Range(H), the (unique) optimal solution w = w(d, h) of minimize w f T w Dw 2, (46) subject to Hw = h, (47) satisfies w M { f T } w + h. (48) The next two results characterize the directions x and ( y, s N ) as optimal solutions of certain convex QP problems. Lemma 7 Suppose that Assumptions 2 and 3 hold. For (x, s, y) IR n ++ IR m, let X diag (x), S diag (s), D X 1/2 S 1/2 and ( x, s, y) denote the solution of (8)(10). Then, (u, v) =( y, s N ) solves the problem 1 min u,v 2 D N v 2, (49) s.t. A T Bu = rb D + Q B x s B, (50) A T N u + v = rn D + Q N x, (51) where Q 2 f(x), and r D f(x)+a T y+s. Proof. By (10), we see that (u, v) =( y, s N ) satisfies the constraints (50) and (51). The result follows once we verify that (u, v) =( y, s N ) satisfies the KKT (Karush KuhnTucker) conditions for the above problem, namely: A N D 2 N s N Range(A B ). (52) Indeed, by (8), we have D 2 s = (x + x). (53) By the definition of (B,N),wehavex N =0where x is as in Assumption 3. This implies that b = Ax = A B x B. Using this fact together with (9) and (53), we obtain A N D 2 N s N = A N (x N + x N ) = A B (x B + x B ) b = A B (x B + x B x B) Range(A B ).
13 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 257 Lemma 8 Suppose that Assumption 2 holds. For (x, s, y) IR n ++ IR m, let X diag (x), S diag (s), D X 1/2 S 1/2 and ( x, s, y) denote the solution of (8)(10). Then, w = xsolves the problem minimize w f(x) T w wt 2 f(x)w D 1 w 2, subject to Aw = b Ax. (54) Proof. Clearly, x is feasible to (54), due to (9). In view of Assumption 2, it remains to verify that x satisfies the first order necessary condition for optimality of (54), namely f(x) + 2 f(x) x + D 2 x Range(A T ). Indeed, by (8), we have D 2 x = (s + s), which together with (10) imply f(x)+ 2 f(x) x + D 2 x = f(x)+ 2 f(x) x s s = A T (y + y) Range(A T ). The following result plays a crucial role in bounding the x B and s N components of the primaldual affine scaling direction. Lemma 9 Suppose that Assumptions 2, 3 and 4 hold. For (x, s, y) IR n ++ IR m, let X diag (x), S diag (s), D X 1/2 S 1/2 and ( x, s, y) denote the solution of (8)(10). Let x = x A + x denote the decomposition of x according to (7). Then there exists a constant C 4 > 0 such that. x C 4 ( x + r + x N ). (55) Proof. Let E be a matrix such that Null (E) =A. In view of Lemma 8 and the fact that E x = E x, it follows that x solves the problem min { f(x) T w + 12 wt 2 f(x)w + 12 } D 1 w 2 Aw = b Ax, Ew = E x. We will now simplify the objective function of (56). Indeed, let w be a feasible solution of (56). Using the fact that w x Null (E) =A Null ( 2 f(x)), wehave w T 2 f(x)w=( x ) T 2 f(x) x. Let (x,s,y )denote the point as in Assumption 3. For every d A, applying the mean value theorem to the function λ f(x + λ(x x )) T d, and using Assumption 4, we conclude that f(x) T d f(x ) T d = (x x ) T 2 f(x +ξ(x x ))d =0, where ξ [0, 1]. Since w x A,we conclude that f(x) T (w x )= f(x ) T (w x ), and hence, that the quantity f(x) T w f(x ) T wis independent of the feasible solution w considered. The above observations yield (56)
14 258 MONTEIRO AND ZHOU x = argmin { f(x ) T w + 12 } D 1 w 2 Aw = b Ax, Ew = E x. (57) Observing that f(x ) T w =(A T y +s ) T w=s T w+y T Aw = s T w + y T (b Ax), (58) for every feasible solution w of (57), we obtain x = argmin {s T w + 12 } D 1 w 2 Aw = b Ax, Ew = E x. (59) Applying Lemma 6 to the above problem, we conclude that there exists L 1 > 0 such that ) x L 1 ( s T x + b Ax + E x ) = L 1 ( s T N x N + r + E x, which yields (55) with C 4 L 1 max{ s N, 1, E }. We are now ready to state the main result of this paper. Theorem 2 Suppose that Assumptions 1, 2, 3 and 4 hold and let X IR n +be a set such that sup { 2 f(x) : x X}<, (60) { d T 2 } f(x)d inf d 2 : x X,d Null (A), d 0 >0. (61) Let a point (x 0,s 0,y 0 ) IR 2n ++ IR m and parameters δ 1,δ 2,η 2 0, and η 1 > 0 be given. Then, there exists a constant C 5 0 with the following property: for any (x, s, y) N (δ 1,η 1,δ 2,η 2 )and t [0, 1/2] satisfying (16) and (17) and the conditions x T s min(1,x 0T s 0 ) and x X, the corresponding solution ( x, s, y) of (8)(10) satisfies max{ x, s } C 5 (x T s) 1 δ, (62) where δ δ 1 +2δ 2. Proof. We only give the proof for the case in which r 0 0, where r 0 is given by (37). Let L 1 and L 2 denote the supremum and infimum in (60) and (61), respectively. Using relations (16) and (17), the fact that x T s 1, δ 2 δ and (x, s, y) N(δ 1,η 1,δ 2,η 2 ), the definition of v(x) given in Lemma 3, the definition of L 1 and Lemma 5, we obtain t = r / r 0 L 3 (x T s) 1 δ2 L 3 (x T s) 1 δ, (63) v(x) = 2 f(x)u 0 + r 0 L 1 u 0 + r 0 L 4 (64) max ( x N, s B ) (C 2 + C 3 L 1 )(x T s) 1 δ = L 5 (x T s) 1 δ, (65)
15 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 259 where L 3 η 2 / r 0 and L 5 C 2 + C 3 L 1. By (63), (64) and (65), we have ( x + tu 0 ) T ( s + tv(x)) = x T B s B + x T N s N +t 2 v(x) T u 0 [ ] +t v(x) T x+u 0 BT sb +u 0 T N sn x B s B + x N s N +t 2 v(x) u 0 +t [ v(x) x + u 0 B s B + u 0 N s N ] L 5 x (x T s) 1 δ +L 5 s N (x T s) 1 δ +L 4 L 2 3 u 0 (x T s) 2 2δ +L 3 (x T s) 1 δ [ L 4 x +L 5 u 0 (x T s) 1 δ + u 0 s N ] L 6 (x T s) 1 δ max { x, s N, (x T s) 1 δ}, (66) where L 6 2L 5 + L 2 3L 4 u 0 + L 3 L 4 + L 3 L 5 u 0 + L 3 u 0. Using relations (26) and (66) and the fact that ( x + tu 0 ) Null (A) and L 2 is the infimum in (61), we obtain L 2 ( x + tu 0 ) 2 ( x + tu 0 ) T 2 f(x)( x + tu 0 ) = ( x+tu 0 ) T ( s + tv(x)) L 6 (x T s) 1 δ max { x, s N, (x T s) 1 δ}. (67) By Lemmas 6 and 7, relations (63) and (65) and the definition of L 1, there exists a constant L 7 > 0 such that ( s N L 7 2 f(x) x + s B + r ) { L 7 L1 x + L 5 (x T s) 1 δ + L 3 r 0 (x T s) 1 δ} L 8 max { (x T s) 1 δ, x }, (68) where L 8 L 7 (L 1 + L 5 + L 3 r 0 ). Combining (67) and (68), it is easy to see that there exists L 9 > 0 such that Thus ( x + tu 0 ) 2 L 9 (x T s) 1 δ max{(x T s) 1 δ, x } (69) x ( x + tu 0 ) + t (u 0 ) ( x + tu 0 ) + L 3 u 0 (x T s) 1 δ L 10 (x T s) (1 δ)/2 max{(x T s) (1 δ)/2, x 1/2 }, where L 10 L 9 + L 3 u 0. This relation together with (55), (63) and (65) imply ( x C 4 x + r + x N ) ( C 4 x +L 3 r 0 (x T s) 1 δ2 +L 5 (x T s) 1 δ) ( C 4 x +L 3 r 0 (x T s) 1 δ +L 5 (x T s) 1 δ) L 11 (x T s) (1 δ)/2 max {(x T s) (1 δ)/2, x 1/2}
16 260 MONTEIRO AND ZHOU where L 11 C 4 (L 5 + L 10 + L 3 r 0 ). It is easy to see that the last relation implies that x L 12 (x T s) 1 δ, where L 12 = max{l 2 11, 1}. This relation together with (65) and (68) clearly imply that (62) holds for some constant C 5 0. Observe that Theorem 1 is an immediate consequence of Theorem 2. Indeed, when the set X is bounded, conditions (60) and (61) follows from Assumptions 2 and Appendix In this appendix, we give the proof of Proposition 1. Proof of Proposition 1: We first prove (a). For any k 1, setting (x, s) =(x k,s k )in (22) and using the fact that t (0, 1), t k [0, t], (x 0,s 0 ) > 0, (x k,s k ) > 0, ( x, s) > 0 and [ f(x 0 ) f( x)] T (x 0 x) 0, we obtain (1 t k )( x T s k + x kt s) t 2 kx 0T s 0 +(1 t k ) 2 x T s+x kt s k +t k (1 t k ) (x 0T s + x T s 0) +t k (1 t k ) [ f(x 0 ) f( x) ] T (x 0 x) x 0T s 0 +(1 t k ) x T s + x kt s k +(1 t k ) (x 0T s+ x T s 0) +(1 t k ) [ f(x 0 ) f( x) ] T (x 0 x). The above inequality, together with the fact that 1 t k 1 t >0, implies x T s k + x kt s 1 ( x 1 t 0T s 0 + x kt s k) + x T s+x 0T s+ x T s 0 + [ f(x 0 ) f( x) ] T (x 0 x). Since ( x, s) > 0, the boundedness of the sequence {(x k,s k )} follows immediately from the boundedness of the sequence {x kt s k }. We next prove (b). Let (ˆx, ŝ, ŷ) be the solution of (2) (6). For any k 1, setting (x, s) =(x k,s k ) in (22) and using the fact that t (0, 1), t k [0, t], (x 0,s 0 ) > 0, (x k,s k )>0,(ˆx, ŝ) 0, ˆx T ŝ =0and [ f(x 0 ) f(ˆx)] T (x 0 ˆx) 0, we obtain t k (x 0T s k + x kt s 0 ) t 2 kx 0T s 0 +(1 t k ) 2ˆx T ŝ+x kt s k +t k (1 t k ) (x 0T ŝ +ˆx T s 0) +t k (1 t k ) [ f(x 0 ) f(ˆx) ] T (x 0 ˆx) t k x 0T s 0 + x kt s k + t k ( x 0T ŝ +ˆx T s 0) +t k [ f(x 0 ) f(ˆx) ] T (x 0 ˆx).
17 SUPERLINEAR CONVERGENCE OF INFEASIBLEINTERIORPOINT 261 Using (19) and the fact that r k = t k r 0,wehavex kt s k /t k x 0T s 0 /ρ, for all k 1. This observation together with the last inequality imply x 0T s k + x kt s 0 x 0T s 0 + xkt s k + x 0T ŝ +ˆx T s 0 t k + [ f(x 0 ) f(ˆx) ] T (x 0 ˆx) x 0T s 0 + x0t s 0 + x 0T ŝ +ˆx T s 0 + ρ [ f(x 0 ) f(ˆx) ] T (x 0 ˆx), (70) Since (x 0,s 0 ) > 0 and the right hand side of (70) does not depend on k, it follows from (70) that {(x k,s k )}is bounded. Using Assumption 1, it is now easy to see that {y k } is also bounded. References 1. C. C. Gonzaga and R. Tapia, On the convergence of the MizunoToddYe algorithm to the analytic center of the solution set, SIAM J. Optim., 7, 1997, pp M. Kojima, N. Megiddo and S. Mizuno, A primaldual infeasibleinteriorpoint algorithm for linear programming, Mathematical Programming, 61, 1993, pp M. Kojima, N. Megiddo and T. Noma, Homotopy continuation methods for nonlinear complementarity problems, Mathematics of Operations Research, 16, 1991, pp S. Mehrotra, Quadratic convergence in a primaldual method, Mathematics of Operations Research, 18, 1993, pp S. Mizuno, Polynomiality of infeasibleinteriorpoint algorithms for linear programming, Mathematical Programming, 67, 1994, pp R. D. C. Monteiro, T. Tsuchiya and Y. Wang, A Simplified Global Convergence Proof of the Affine Scaling Algorithm, Annals of Operations Research, 47, 1993, R. D. C. Monteiro and Y. Wang, Trust region affine scaling algorithms for linearly constrained convex and concave programs, Technical Report, School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA, June, To appear in Mathematical Programming. 8. R. D. C. Monteiro and S. Wright, A globally and superlinearly convergent potential reduction interior point method for convex programming, Technical Report, 92 13, Dept. of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA, July, R. D. C. Monteiro and S. Wright, Local convergence of interiorpoint algorithms for degenerate monotone LCPs, Computational Optimization and Applications, 3, 1994, pp R. D. C. Monteiro and S. Wright, Superlinear primaldual affine scaling algorithms for LCP, Mathematical Programming, 69, 1995, pp R. D. C. Monteiro and S. Wright, A superlinear infeasibleinteriorpoint affine scaling algorithm for LCP, SIAM Journal on Optimization, 6, 1996, pp F. A. Potra, An O(nL) infeasibleinteriorpoint algorithm for LCP with quadratic convergence, Ann. of Oper. Res., 62, 1996, pp F. A. Potra, A quadratically convergent predictorcorrector method for solving linear programs from infeasible starting points, Mathematical Programming, 67, 1994, pp F. A. Potra and Y. Ye, Interior point methods for nonlinear complementarity problems, J. Optim. Theory Appl., 88, 1996, pp F. A. Potra and Y. Ye, A quadratically convergent polynomial algorithm for solving entropy optimization problems, SIAM Journal on Optimization, 3, 1993, pp
18 262 MONTEIRO AND ZHOU 16. E. M. Simantiraki and D. F. Shanno, An infeasibleinteriorpoint method for linear complementarity problems, RUTCOR Research Report, RRR 795, Rutgers Center for Operations Research, Rutgers University, New Brunswick, NJ 08903, USA, P. Tseng and Z. Q. Luo, On the convergence of the affine scaling algorithm, Mathematical Programming, 56, 1992, pp Tunçel, L., Constant potential primaldual algorithms: A framework, Mathematical Programming, 66, 1994, pp S. Wright, A pathfollowing infeasibleinteriorpoint algorithm for linear complementarity problems, Optimization Methods and Software, 2, 1993, pp S. Wright, A pathfollowing interiorpoint algorithm for linear and quadratic optimization problems, Ann. Oper. Res., 62, 1996, pp S. Wright, An infeasible interior point algorithm for linear complementarity problems, Mathematical Programming, 67, 1994, S. Wright and D. Ralph, A superlinear infeasibleinteriorpoint algorithm for monotone complementarity problems, Math. Oper. Res., 21, 1996, pp Y. Ye, On the Qorder of convergence of interiorpoint algorithms for linear programming, in Proceedings of the 1992 Symposium on Applied Mathematics, F. Wu, ed., Institute of Applied Mathematics, Chinese Academy of Sciences, 1992, pp Y. Ye and K. Anstreicher, On quadratic and O( nl) convergence of a predictorcorrector algorithm for LCP, Mathematical Programming, 62, 1993, pp Y.Ye,O.Güler, R. A. Tapia and Y. Zhang, A quadratically convergent O( nl) iteration algorithm for linear programming, Mathematical Programming, 59, 1993, pp Y. Zhang, On the convergence of a class of infeasible interior point methods for the horizontal linear complementarity problem, SIAM Journal on Optimization, 4, 1994, pp Y. Zhang and R. A. Tapia, Superlinear and quadratic convergence of primal dual interior point methods for linear programming revisited, Journal of Optimization Theory and Applications, 73, 1992, pp Y. Zhang and R. A. Tapia, A superlinearly convergent polynomial primal dual interior point algorithm for linear programming, SIAM Journal on Optimization, 3, 1993, pp Y. Zhang, R. A. Tapia and J. E. Dennis, On the superlinear and quadratic convergence of primal dual interior point linear programming algorithms, SIAM Journal on Optimization, 2, 1992, pp Y. Zhang and D. Zhang, On polynomiality of the Mehrotratype predictorcorrector interiorpoint algorithms, Mathematical Programming, 68, 1995, pp
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