TY - THES
AU - Boljunčić, Jadranka
PY - 1987
TI - Quadratic programming : quantitative analysis and polynomial running time algorithms
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - Many problems in economics, statistics and numerical analysis can be formulated as the optimization of a convex quadratic function over a polyhedral set. A polynomial algorithm for solving convex quadratic programming problems was first developed by Kozlov at al. (1979). Tardos (1986) was the first to present a polynomial
algorithm for solving linear programming problems in which the number of arithmetic steps depends only on the size of the numbers in the constraint matrix and is independent of the size of the numbers in the right hand side and the cost coefficients. In the first part of the thesis we extended Tardos' results to strictly convex quadratic programming of the form max {cTx-½xTDx : Ax ≤ b, x ≥0} with D being symmetric positive definite matrix. In our algorithm the number of arithmetic steps is independent of c and b but depends on the size of the entries of the matrices A and D.
Another part of the thesis is concerned with proximity and sensitivity of integer and mixed-integer quadratic programs. We have shown that for any optimal solution z̅ for a given separable quadratic integer programming problem there exist an optimal solution x̅ for its continuous relaxation such that
AB - z̅ - x̅
AB - ∞≤n∆(A) where n is the number of variables and ∆(A) is the largest absolute sub-determinant of the integer constraint matrix A . We have further shown that for any feasible solution z, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution z̅ having greater objective function value and with
AB - z - z̅
AB - ∞≤n∆(A). Under some additional assumptions the distance between a pair of optimal solutions to the integer quadratic programming
problem with right hand side vectors b and b', respectively, depends linearly on
AB - b — b'
AB - ₁. The extension to the mixed-integer nonseparable quadratic case is also given.
Some sensitivity analysis results for nonlinear integer programming problems are given. We assume that the nonlinear 0 — 1 problem was solved by implicit enumeration and that some small changes have been made in the right hand side or objective function coefficients. We then established what additional information to keep in the implicit enumeration tree, when solving the original problem, in order to provide us with bounds on the optimal value of a perturbed problem. Also, suppose that after solving the original problem to optimality the problem was enlarged by introducing a new 0 — 1 variable, say xn+1. We determined a lower bound on the added objective function coefficients for which the new integer variable xn+1 remains at zero level in the optimal solution for the modified integer nonlinear program. We discuss the extensions to the mixed-integer case as well as to the case when integer variables are not restricted to be 0 or 1. The computational results for an example with quadratic objective function, linear constraints and 0—1 variables are provided.
Finally, we have shown how to replace the objective function of a quadratic program
with 0—1 variables ( by an integer objective function whose size is polynomially bounded by the number of variables) without changing the set of optimal solutions. This was done by making use of the algorithm given by Frank and Tardos (1985) which in turn uses the simultaneous approximation algorithm of Lenstra, Lenstra and Lovász (1982).
N2 - Many problems in economics, statistics and numerical analysis can be formulated as the optimization of a convex quadratic function over a polyhedral set. A polynomial algorithm for solving convex quadratic programming problems was first developed by Kozlov at al. (1979). Tardos (1986) was the first to present a polynomial
algorithm for solving linear programming problems in which the number of arithmetic steps depends only on the size of the numbers in the constraint matrix and is independent of the size of the numbers in the right hand side and the cost coefficients. In the first part of the thesis we extended Tardos' results to strictly convex quadratic programming of the form max {cTx-½xTDx : Ax ≤ b, x ≥0} with D being symmetric positive definite matrix. In our algorithm the number of arithmetic steps is independent of c and b but depends on the size of the entries of the matrices A and D.
Another part of the thesis is concerned with proximity and sensitivity of integer and mixed-integer quadratic programs. We have shown that for any optimal solution z̅ for a given separable quadratic integer programming problem there exist an optimal solution x̅ for its continuous relaxation such that
N2 - z̅ - x̅
N2 - ∞≤n∆(A) where n is the number of variables and ∆(A) is the largest absolute sub-determinant of the integer constraint matrix A . We have further shown that for any feasible solution z, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution z̅ having greater objective function value and with
N2 - z - z̅
N2 - ∞≤n∆(A). Under some additional assumptions the distance between a pair of optimal solutions to the integer quadratic programming
problem with right hand side vectors b and b', respectively, depends linearly on
N2 - b — b'
N2 - ₁. The extension to the mixed-integer nonseparable quadratic case is also given.
Some sensitivity analysis results for nonlinear integer programming problems are given. We assume that the nonlinear 0 — 1 problem was solved by implicit enumeration and that some small changes have been made in the right hand side or objective function coefficients. We then established what additional information to keep in the implicit enumeration tree, when solving the original problem, in order to provide us with bounds on the optimal value of a perturbed problem. Also, suppose that after solving the original problem to optimality the problem was enlarged by introducing a new 0 — 1 variable, say xn+1. We determined a lower bound on the added objective function coefficients for which the new integer variable xn+1 remains at zero level in the optimal solution for the modified integer nonlinear program. We discuss the extensions to the mixed-integer case as well as to the case when integer variables are not restricted to be 0 or 1. The computational results for an example with quadratic objective function, linear constraints and 0—1 variables are provided.
Finally, we have shown how to replace the objective function of a quadratic program
with 0—1 variables ( by an integer objective function whose size is polynomially bounded by the number of variables) without changing the set of optimal solutions. This was done by making use of the algorithm given by Frank and Tardos (1985) which in turn uses the simultaneous approximation algorithm of Lenstra, Lenstra and Lovász (1982).
UR - https://open.library.ubc.ca/collections/831/items/1.0097502
ER - End of Reference