One-Sided Differentials. Quasidifferential. Necessary and Sufficient Optimality Conditions. Finite-Dimensional Nonsmooth Functions. Subdifferentiable Functions.
3170
Q
Quadratic Semi-assignment Problem
See also ! Complexity Theory: Quadratic Programming ! Quadratic Assignment Problem ! Quadratic Fractional Programming: Dinkelbach Method ! Quadratic Knapsack ! Quadratic Programming with Bound Constraints ! Standard Quadratic Optimization Problems: Algorithms ! Standard Quadratic Optimization Problems: Applications ! Standard Quadratic Optimization Problems: Theory ! Volume Computation for Polytopes: Strategies and Performances References 1. Celis MR, Dennis JE, Tapia RA (1984) A trust region strategy for nonlinear equality constrained optimization. In: Boggs PT Byrd R, Schnable R (ed) Numerical Optimization. SIAM, Philadelphia, pp 71–82 2. Dennis JE Jr, Schnable RE (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Englewood Cliffs 3. Fu M, Luo Z-Q, Ye Y (1996) Approximation algorithms for quadratic programming. J Combin Optim (to appear) 4. Gay DM (1981) Computing optimal locally constrained steps. SIAM J Sci Statist Comput 2:186–197 5. Gibbons LE, Hearn DW, Pardalos PM (1996) A continuous based heuristic for the maximum clique problem. In: DIMACS. Amer. Math. Soc., Providence, pp 103–124 6. Kamath AP, Karmarkar NK, Ramakrishnan KG, Resende MGC (1992) A continuous approach to inductive inference. Math Program 57:215–238 7. Levenberg K (1963) A method for the solution of certain non-linear problems in least squares. Quart Appl Math 2:164–168 8. Marquardt DW (1963) An algorithm for least-squares estimation of nonlinear parameters. J SIAM 11:431–441 9. Martinez JM (1994) Local minimizers of quadratic functions on Euclidean balls and spheres. SIAM J Optim 4:159–176 10. Moré JJ (1977) The Levenberg-Marquardt algorithm: Implementation and theory. In: Watson GA (ed) Numerical Analysis. Springer, Berlin 11. Nesterov YuE, Nemirovskii AS (1993) Interior point polynomial methods in convex programming: Theory and algorithms. SIAM, Philadelphia 12. Pardalos PM, Rosen JB (1987) Constrained global optimization: Algorithms and applications. Lecture Notes Computer Sci, vol 268. Springer, Berlin
13. Powell MJD, Yuan Y (1991) A trust region algorithm for equality constrained optimization. Math Program 49:189– 211 14. Rendl F, Wolkowicz H (1994) A semidefinite framework for trust region subproblems with applications to large scale minimization. CORR Report 94-32, Dept Combinatorics and Optim, Univ Waterloo, Ontario 15. Sorenson DC (1982) Newton’s method with a model trust region modification. SIAM J Numer Anal 19:409–426 16. Vavasis SA (1991) Nonlinear optimization: Complexity issues. Oxford Sci. Publ., Oxford 17. Vavasis SA (1993) Polynomial time weak approximation algorithms for quadratic programming. In: Pardalos PM (ed) Complexity in Numerical Optimization. World Sci., Singapore 18. Vavasis SA, Zippel R (1990) Proving polynomial time for sphere-constrained quadratic programming. Techn Report 90–1182, Dept Computer Sci, Cornell Univ, Ithaca, NY 19. Ye Y (1992) On affine scaling algorithms for nonconvex quadratic programming. Math Program 56:285–300 20. Ye Y (1994) Combining binary search and Newton’s method to compute real roots for a class of real functions. J Complexity 10:271–280 21. Yuan Y (1990) On a subproblem of trust region algorithms for constrained optimization. Math Program 47:53–63
Quadratic Semi-assignment Problem QSAP LEONIDAS PITSOULIS Princeton University, Princeton, USA MSC2000: 90C27, 90C11, 90C08 Article Outline Keywords See also References Keywords Optimization Consider that we have n ‘objects’ and m ‘locations’, n > m, and we want to assign all objects to locations with at least one object to each location, so as to minimize the overall distance covered by the flow of materials moving between different objects. Given a flow matrix F =
Quasidifferentiable Optimization
(f ij ) and a distance matrix D = (dij ), we can formulate the quadratic semi-assignment problem as follows: 8 ˆ ˆ ˆ min ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ s.t. ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :
n m!1 n X m X X X
f i j d k l x ki x l j
iD1 jD1 kD1 l DkC1 n m X X
C
n X jD1
bi j xi j
iD1 jD1
x i j D 1;
x i j 2 f0; 1g;
i D 1; : : : ; m;
i D 1; : : : ; m;
j D 1; : : : ; n:
Comparing the above formulation with that of the quadratic assignment problem (cf. ! Quadratic assignment problem), we can see that the QSAP is a relaxed version of the QAP, where instead of assignment constraints we have semi-assignment constraints. SQAP unifies some interesting combinatorial optimization problems like clustering and m-coloring. In a clustering problem we are given n objects and a dissimilarity matrix F = (f ij ). The goal is to find a partition of these objects into m classes so as to minimize the sum of dissimilarities of objects belonging to the same class. Obviously this problem is a QSAP with coefficient matrices F and D, where D is an m × m identity matrix. In the m-coloring problem we are given a graph with n vertices and want to check whether its vertices can be colored by m different colors such that each two vertices which are joined by an edge receive different colors. This problem can be modeled as a SQAP with F equal to the adjacency matrix of the given graph and D the m × m identity matrix. The m-coloring has an answer ‘yes’ if and only if the above SQAP has optimal value equal to 0. Practical applications of the SQAP include distributed computing [5] and scheduling [1]. SQAP was originally introduced by D.E. Greenberg [2]. As pointed out in [3], this problem is NP-hard. I.Z. Milis and V.F. Magirou [5] propose a Lagrangian relaxation algorithm for this problem, and show that similarly as for for the QAP, it is very hard to provide optimal solutions even for SQAPs of small size. Lower bounds for the SQAP have been provided in [4], and polynomially solvable special cases have been discussed in [3].
Q
See also ! Feedback Set Problems ! Generalized Assignment Problem ! Graph Coloring ! Graph Planarization ! Greedy Randomized Adaptive Search Procedures ! Quadratic Assignment Problem References 1. Chretienne P (1989) A polynomial algorithm to optimally schedule tasks on a virtual distributed system under treelike precedence constraints. Europ J Oper Res 43:225– 230 2. Greenberg DE (1969) A quadratic assignment problem without column constraints. Naval Res Logist Quart 16:417– 422 3. Malucelli F (1993) Quadratic assignment problems: solution methods and applications. PhD Thesis, Dip. Informatica, Univ. Pisa 4. Malucelli F, Pretolani D (1993) Lower bounds for the quadratic semi-assignment problem. Techn Report 955, Centre Rech Transports, Univ Montréal, Canada 5. Milis IZ, Magirou VF (1995) A Lagrangean relaxation algorithm for sparse quadratic assignment problems. Oper Res Lett 17:69–76 6. Stone HS (1977) Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans Softw Eng 4:85–93
Quasidifferentiable Optimization GEORGIOS E. STAVROULAKIS Carolo Wilhelmina Techn. University, Braunschweig, Germany MSC2000: 49J52, 26B25, 90C99, 26E25 Article Outline Keywords One-Dimensional Nonsmooth Functions
One-Sided Differentials Quasidifferential Necessary and Sufficient Optimality Conditions
Finite-Dimensional Nonsmooth Functions Subdifferentiable Functions Superdifferentiable Functions Quasidifferentiable Functions
Further Related Topics
3171
