Solving Obstacle Problems by Using a New Interior Point Algorithm. Abstract

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1 Solving Obstacle Problems by Using a New Interior Point Algorithm Yi-Chih Hsieh Department of Industrial Engineering National Yunlin Polytechnic Institute Huwei, Yunlin 6308 Taiwan and Dennis L. Bricer Department of Industrial Engineering University of Iowa Iowa City, IA 54 USA (August, 996) Abstract A new infeasible path-following algorithm, which follows a path on the complementarity surface, is proposed for solving obstacle problems. The sequence of iterates generated by the algorithm does not satisfies the primal-dual feasibilities as do the other path-following algorithms which follow the central path, but satisfies the complementarity equations at each iteration. Numerical results show that only few number of iterations are required for solving such class of problems. Keywords: Infeasible Interior Point algorithm, Obstacle Problems, Quadratic Programming.

2 . Introduction Obstacle problems are typical problems in engineering and Physics. Several problems, such as the lubrication problem, the elastoplastic torsion problem and Signorini problem etc., are all the applied obstacle problems (Rodrigues (987)). For obstacle problems, it is assumed that a homogeneous membrane occupying a domain D of the Oxy plane is equally stretched in all direction by a uniform tension and loaded by a normal uniformly distributed force ρ, and it is also assumed that each point (x,y) of the membrane is displaced by an amount v( x, y) vertically to the plane Oxy. One deforms the boundary D of the membrane conformly by prescribing its displacement g = g( x, y), that is, v = g on D. Suppose that the potential energy of the deformed is proportional to the increase in the area of its surface. Since the area is + v x + v D y dxdy + v ( x + v y ) dxdy, D the change in the area of the membrane is equal to v ( x + v ) D y dxdy = D v dd. The potential energy of deformation has the following expression d(v) = λ and without loss of generality, it is set λ =. D v dd, Since the wor done by the external forces during the actual displacement is given by thus the total potential energy is given by e(v) = D ρvdd, d(v) e(v) = D v dd ρv dd. (.) D The obstacle problem is to determine the surface of an elastic membrane subject to a vertical force with upper and lower bounds of the surface. More explicitly speaing, it is to find the

3 equilibrium position, with minimal total potential energy in (.), of an elastic membrane subject to the following restrictions:. the membrane is subject to the action of a vertical force,. the membrane must lie over an "obstacle" (constraints) within the upper and lower bounds. The mathematical formulation for the obstacle problem is expressed by Min q(v) = D v dd ρvdd D s.t. v { v H 0 (D) : l v u on D} (.) where ρ is a force function, D is a bounded open set, and H 0 (D) is a space of all functions with compact support in D such that v and v belong to L (D) (see Ciarlet (978) for details). Several researchers have numerically solved the obstacle problems with various approaches. Moré and Toraldo (99) proposed an algorithm that uses the conjugate gradient method to explore the face of the feasible region defined by the current iterate, and the gradient projection method to move to a different face. The algorithm is limited to problems with bound constraints. Contrarily, Han et al. (99) implement the potential reduction algorithm (an interior point algorithm) to solve the obstacle problem using a given feasible initial solution. In this paper, a new infeasible path-following algorithm in which the initial solution is positive, infeasible, and exactly on the complementarity surface is used to solve the obstacle problem. As numerical results shown, this approach can find solution with the use of only few iterations.. The New Infeasible Interior Point Algorithm In this paper, obstacle problems are discretized into convex quadratic programming and then solved by our new infeasible interior point algorithm. The procedure of this algorithm for general convex quadratic programming is derived in this section, and the convergence of this algorithm can be found in Hsieh and Bricer (996). Consider the convex quadratic programming problems with linear constraints in the following standard form:

4 (QP) Min s.t. xt Qx + c T x Ax y = b x, y 0 Its dual is: (QPD) Max xt Qx + b T w s.t. Qx + A T w + s = c s,w 0 where x, s,c R n, y,w, b R m, Q R n n, and A R m n. The following assumptions are also imposed. (A) The matrix Q is positive semi-definite. (A) The constraint matrix A has full row ran. (A3) The feasible region is nonempty and bounded. For x, y > 0 in (QP) and s, w > 0 in (QPD), we can apply the logarithmic barrier function technique, and obtain the nonlinear programming problems, (QP µ ) and (QPD µ ): n m (QPµ) Min xt Qx + c T x µ log x j µ log y j s.t. Ax y = b x, y > 0 j = j = and (QPDµ) Max xt Qx + b T w + µ log w j + µ log s j s.t. Qx + A T w + s = c w,s > 0 m j = n j = where µ > 0 is a barrier parameter. It is expected that the optimal solution of problem (QP µ ) would converge to the optimal solution of the original problem (QP) as µ 0. Convex programming theory further implies that the global solution, if one exists, is completely characterized by the KKT conditions as: Ax y = b x, y > 0 (primal feasibility) (.a) Qx + A T w + s = c s,w > 0 (dual feasibility) (.b)

5 XSe n = µe n (complementary slacness) (.c) WYe m = µe m (complementary slacness) (.d) where X,S,W, and Y are diagonal matrices with diagonal entries equal to the components of x, s, w, and y, respectively, and e i is the column matrix with i elements, each with value one. Assume that (x,y,s,w ) >0 is a current solution of equation (.) for given µ >0. Let x = e z x, y = e z y, s = e z s, and w = e z w. (.) Substituting (.) to (.) and applying Newton's method, we have AX Y 0 0 QX 0 S A T W X S 0 X S 0 0 W Y 0 W Y dz x dz y dz s dz w η (Ax y b) η (Qx + A T w + s c) = X S e n µ e = n W Y e m µ e m t t t 3 t 4 (.3) where 0 < η <, which is used to control the norm of the direction vector. Therefore, after solving (.3), a new solution can be obtained by step size one (normal step size) such that x + = x e dz x, y + = y e dz y, s + = s e dz s, and w + = w e dz w (.4) For each iteration, the barrier parameter µ is adjusted as follows: µ = σ (x ) T s + (y ) T w n + m, where 0 < σ <. (.5) Next, we introduce the infeasible interior point algorithm for QP problems. Infeasible Interior Point Algorithm. Step :(Initialization) Set =0. Choose three small values for ε, ε, and ε 3, respectively. Define σ 0 and η 0. Start with any initial solution (x 0,y 0,s 0,w 0 )>0 which satisfy X 0 S 0 e n = e n and W 0 Y 0 e m = e m. Step : (Intermediate computation) Compute µ by (.5) and t,t,t 3 and t 4 by (.3), respectively. Step 3 : (Checing optimality) If µ < ε, t b + < ε, and t Qx + c + < ε 3, then stop; the current solution is accepted as the optimal solution. Else proceed to the next step. Step 4 : (Finding the directions) Compute dz w,dz x,dz y, and dz s by (.3).

6 Step 5 : (Finding the new iterate) Compute x +, y +,s + and w + by (.4). Set = + and go to step. It should be noted that the sequence generated by the algorithm follows a path on the complementarity surface, that is, (.c) and (.d) are satisfied for each iteration, which is different from the central path ((.a) and (.b) are satisfied for each iteration) followed by most of infeasible path-following algorithms (Renegar (988), Monteiro and Adler (989a, 989b), and Monteiro, Adler and Resende (990) etc.), with the use of Newton's method for solving (.). One should be also noted that if Monomial method is used to solve (.), the sequence generated follows a similar path on the complementarity surface as that of by our new algorithm (Hsieh and Bricer (996)). 3. Discretization and Numerical Results After finite approximation, the obstacle problem is reduced to a QP problem with box constraints. Let D=[a,a ] [b,b ] be a rectangle in R. To discretize, we draw m x equally spaced vertical lines perpendicular to the x axis between [a,a ] and m y equally spaced vertical lines perpendicular to the y axis between [b,b ]. The grid points formed by the intersection of these lines are labeled to v i, j. For simplicity, assume that the force function ρ is constant. Then the objective function of the obstacle problem (.) becomes q(v) = 4 m x m y q i, j i = j = m x m y ρh x h y v i, j i= j = where v i+, j v i, j q i, j (v) = h x h y h x + v v i, j + i, j h y + v i,j v i, j h x + v v i, j i, j, h y h x = a a m x +, and h y = b b m y +. If we further assume that m x = m y = m and assign a variable x i, j to each grid point v i, j, then the matrix Q for problem (QP) may be expressed as a symmetric positive definite matrix

7 P I I P I 0 I P I Q = R n n, I P I I P where P = R m m, I R m m is the identity matrix, and n = m. The vector c for problem (QP) is given by c i = ρh, i =,,..., n, where h = m +. Hence the obstacle problem in (.) can be discretized by a quadratic programming (QP). In this paper, two obstacle problems with characteristics in Table are solved by the new pathfollowing algorithm. All the numerical results are obtained by using HP75/75 worstation with program coded in APL. Table The characteristics for the obstacle problems. Obstacle Prob. Problem (I) Problem (II) Force ρ = ρ = Lower bounds ( ( ) sin( 9.3γ i )) l i = sin( 3.α i ) sin( 3.3γ i ) l i = sin 9.α i Note: For these two obstacle problems, α i = i i m m for i =,,..., n. ( ) h and γ i = i m h

8 We apply the algorithm to solve these two obstacle problems under the following conditions. (C) n = 65. That is, there are 65 primal variables for the problem, and the total number of variables for the KKT equations is,500. (C) The initial iterate is x 0 = y 0 = (0,...,0) t R n and s 0 = w 0 = (0.0,...,0.0) t R n is infeasible for (.a) and (.b), and is exactly on a path of the complementarity surface. (C3) σ 0 = 0.5. (C4) ε = ε = ε 3 = (C5) η 0 =0.95 and η + = max(0., 0.5η ) if max(dzx i,dzs i, dzy j,dzw j ) > 5 i, j. min(0.95,.η ) otherwise The results are illustrated in Figures -0. From these figures we observe that. For Problem (I), figure of the obstacle based upon the constraints in Table is depicted in Figure, and the initial iterate is depicted in Figure which is a plane, since all initial solutions have value 0. Figure 3, for iteration 5, shows that there is a trend for the plane to project upwards for several points near the boundary. We also find that Figure 4 (at iteration 0) is very similar to Figure 5, the final solution (at iteration 0).. The residuals of the primal-dual feasibility and complementarity for the Problem (I) are illustrated in Figure 6. From this figure we observe that the residual for the complementary slacness is reduced faster than that of primal feasibility and dual feasibility. 3. For Problem (II), we observe the similar property. After 0 iterations, the solution, Figure 7, is extremely close to the final solution (at iteration 5), Figure The total number of iterations needed for these two problems are 0 and 5, respectively, which are very small even though the total number of variables is up to, Conclusion Obstacle problems have been widely investigated and have various applications in the real world. In this paper, after discretization of the obstacle problem into a quadratic programming, a new

9 infeasible interior point algorithm, which follows a path on the complementarity surface, is used to solve such obstacle problems. Limited numerical results show that this new path-following algorithm can efficiently solve large scale quadratic problems with requiring only few iterations. References Ciarlet, P.G. (978) The finite element method elliptic problems. (Studies in Mathematics and Its Application, Vol. 4, edited by Linos, J.L.), North-Holland, NY. Han, C.G., Pardalos, P.M. and Ye, Y. (99) Solving some engineering problems using an interiorpoint algorithm, woring paper, Pennsylvania State University, University Par, PA. Hsieh, Yi-Chih, and Dennis L. Bricer (996) New infeasible interior point algorithm based on Monomial method", Computers & Operations Research, 3, Hsieh, Y.C. and Bricer, D.L. (996) Infeasible interior following a path on the complementarity surface, woring paper (submitted), University of Iowa, IA. Monteiro, R.D.C. and Adler, I. (989a) Interior path following primal-dual algorithms. Part I: linear programming, Mathematical Programming, 44, 7-4. Monteiro, R.D.C. and Adler, I. (989b) Interior path following primal-dual algorithms. Part II: convex quadratic programming, Mathematical Programming, 44, Monteiro, R.D.C., Alder, I. and Resende, M. (990) A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power extension, Mathematics of Operations Research, 5, 9-4. Moré, J.J. and Toraldo, G. (99) On the solution of large quadratic programming problems with bound constraints, SIAM Journal on Optimization,, Rodrigues, J.J. (987) Obstacle problems in mathematical Physics. (North-Holland Mathematics Studies, Vol. 34, edited by Nachbin, L.), North-Holland, NY.

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