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1.1 the Linear Complementarity Problem and Its Geometry

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1.1 the Linear Complementarity Problem and Its Geometry Chapter LINEAR COMPLEMENTARITY PROBLEM ITS GEOMETRY AND APPLICATIONS THE LINEAR COMPLEMENTARITY PROBLEM AND ITS GEOMETRY The Linear Complementarity Problem abbreviated as LCP is a general problem which unies linear and quadratic programs and bimatrix games The study of LCP has led to many far reaching b enets For example an algorithm known as the com plementary pivot algorithm rst develop ed for solving LCPs has b een generalized in a direct manner to yield ecient algorithms for computing Brouwer and Kakutani xed p oints for computing economic equilibria and for solving systems of nonlinear equations and nonlinear programming problems Also iterative metho ds develop ed for solving LCPs hold great promise for handling very large scale linear programs which cannot b e tackled with the well known simplex metho d b ecause of their large size and the consequent numerical diculties For these reasons the study of LCP oers rich rewards for p eople learning or doing research in optimization or engaged in practical applications of optimization In this book we discuss the LCP in all its depth n Let M beagiven square matrix of order n and q a column vector in R Through out this b o ok we will use the symbols w w z z to denote the variables in n n the problem In an LCP there is no ob jective function to be optimized The T T problem is nd w w w z z z satisfying n n w Mz q w z and w z for all i i i Chapter Linear Complementarity Problem Its Geometry and Applications The only data in the problem is the column vector q and the square matrix M So we n n will denote the LCP of nding w R z R satisfying by the symbol q M It is said to be an LCP of order n In an LCP of order n there are n variables As a sp ecic example let n M q This leads to the LCP w z z w z z w w z z and w z w z The problem can be expressed in the form of a vector equation as w w z z w w z z and w z w z In any solution satisfying at least one of the variables in each pair w z j j has to equal zero One approach for solving this problem is to pick one variable from each of the pairs w z w z and to x them at zero value in The remaining variables in the system may be called usable variables After eliminating the zero variables from if the remaining system has a solution in which the usable variables are nonnegative that would provide a solution to and Pick w w as the zerovalued variables After setting w w equal to in the remaining system is q z q z q z z q2 q1 -2 ()-1 -1 (-2 ) Figure A Complementary Cone Equation has a solution i the vector q can be expressed as a nonnegative T T linear combination of the vectors and The set of all nonnegative The Linear Complementarity Problem and its Geometry T T linear combinations of and is a cone in the q q space as in T Figure Only if the given vector q lies in this cone do es the LCP have a solution in which the usable variables are z z We verify that the p oint T do es lie in the cone that the solution of is z z and hence a solution for is w w z z The cone in Figure is known as a complementary cone asso ciated with the LCP Complementary cones are generalizations of the wellknown class of quadrants or orthants Notation The symbol I usually denotes the unit matrix If we want to emphasize its order we denote the unit matrix of order n b y the symbol I n We will use the abbreviation LP for Linear Program and BFS for Basic Feasible Solution See LCP is the abbreviation for Linear Complementarity Problem and NLP is the abbreviation for Nonlinear Program Column and Row Vectors of a Matrix If A a is a matrix of order m n say we will denote its j th column vector ij T a a by the symbol A and its ith row vector a a by A j mj j i in i Nonnegative Semip ositive Positive Vectors n T Let x x x R x that is nonnegative if x for all j Clearly n j x is said to be semip ositive denoted by x if x for all j and at least j one x Notice the distinction in the symbols for denoting nonnegative with j two lines under the and semip ositive with only a single line under the the zero vector is the only nonnegativevector which is not semip ositive Also if x P n x The vector x strictly p ositive if x for all j Given twovectors j j j n x y R we write x y if x y x y if x y and xy if x y Pos Cones n r r If fx x g R the cone fx x x x g is denoted r r r by Posfx x g Given the matrix A of order m n PosA denotes the cone T PosfA A g fx x A for g n n Directions Rays HalfLines and Step Length n n R Given the direction y its ray Any p oint y R y denes a direction in is the halfline obtained by joining the origin to y and continuing indenitely in the Chapter Linear Complementarity Problem Its Geometry and Applications n same direction it is the set of p oints fy g Given x R by moving from x in the direction y we get p oints of the form x y where and the set of all such p oints f x y g is the haline or ray through x in the direction y The p oint x y for issaidtohave b een obtained bymoving from x in the direction n T y a step length of As an example if y R the ray of y is the set of all T T p oints of the form f g In addition if x the haline through T x in the direction y is the set of all p oints of the form f g See T Figure In this halfline letting we get the p oint and this p oint is T T x in the direction y obtained by taking a step of length from x y y Ray of Half-line or ray through y in the direction of x Figure Rays and HalfLines Complementary Cones In the LCP q M the complementary cones are dened by the matrix M The p oint q do es not play any role in the denition of complementary cones Let M be a given square matrix of order n For obtaining C M the class of complementary cones corresp onding to M the pair of column vectors I M is j j The Linear Complementarity Problem and its Geometry known as the j th complementary pair of vectors j n Pick a vector from the pair I M and denote it by A The ordered set of vectors A A is j j j n known as a complementary set of vectors The cone Pos A A fy y n A A g is known as a complementary cone in the n n n n class C M Clearly there are complementary cones Example Let n andM I In this case the class C I is just the class of orthants in R In n general for any n C I is the class of orthants in R Thus the class of complementary cones is a generalization of the class of orthants See Figure Figures and provide some more examples of complementary cones In the example in Figure since fI M g is a linearly dep endent set the cone PosI M has an empty consists of all the p oints on the horizontal axis in Figure the thick interior It axis The remaining three complementary cones have nonemptyinteriors Pos( I 1 , I 2 ) Pos( I 1 , I 2 ) Pos( M 1, I 2 ) Pos( I 1 , I 2 ) I 2 I 2 I 1 I I 1 1 , I 2 M 1 Pos( I 1 M 2 ) Pos( I , I ) Pos( I , I ) 1 2 1 2 M 2 Pos( M 1 , M 2 ) Figure When M I the Complementarity Cones are the Orthants Figure Complementary Cones when M Chapter Linear Complementarity Problem Its Geometry and Applications Degenerate Nondegenerate Complementary Cones Let PosA A b e a complementary cone in C M This cone is said to b e a non n degenerate complementary cone if it has a nonemptyinterior that is if fA A g n is a linearly indep endent set degenerate complementary cone if its interior is empty which happ ens when fA A g is a linearly dep endent set As examples all the n complementary cones in Figures are nondegenerate In Figure the complementary cone PosI is degenerate the remaining three complemen M tary cones are nondegenerate M 2 I 2 I 2 M 2 I 1 I 1 M 1 M 1 Figure Complementary Cones when M Figure Complementary Cones when M The Linear Complementary Problem n Given the square matrix M of order n and the column vector q R theLCPq M is equivalent to the problem of nding a complementary cone in C M that contains the p oint q that is to nd a complementary set of column vectors A A such n that M g for j n i A fI j j j ii q can be expressed as a nonnegative linear combination of A A n The Linear Complementarity Problem and its Geometry where I is the identity matrix of order n and I is its j th column vector This is j P P n n n n equivalent to nding w R z R satisfying I w M z q w j j j j j j j z for all j and either w or z for all j In matrix notation this is j j j w Mz q w z w z for all j j j P n T Because of the condition is equivalent to w z w z this con j j j dition is known as the complementarity constraint In any solution of the LCP q M if one of the variables in the pair w z is p ositive the other should be zero j j complementary pair of variables and Hence the pair w z is known as the j th j j eachvariable in this pair is the complement of the other In the column vector corresp onding to w is I and the column vector corresp onding to z is M For j j j j j to n the pair I M is the j th complementary pair of column vectors in j j be the the LCP q M For j to n let y fw z g and let A column vector j j j j corresp onding to y in So A fI M g Then y y y is a com j j j j n plementary vector of variables in this
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