Ipsol: an interior point solver for nonconvex optimization problems

• Authors: Kaustuv
• Venue: Ipsol: an interior point solver for nonconvex optimization problems
• Year: 2009

For years, the Systems Optimization Laboratory (SOL) has been a center for research on large-scale nonlinear optimization. This research has given rise to reference solvers MINOS, NPSOL and SNOPT. In this thesis we present IPSOL, a prototype solver for general nonlinear optimization problem. IPSOL differs from these reference solvers in two ways. First, it uses second-derivative information, and second, it uses a barrier formulation to handle inequality constraints. Together, these features enable IPSOL to solve large-scale optimization problems efficiently. IPSOL solves a sequence of equality-constrained log-barrier subproblems with decreasing values μ until convergence is attained. The subproblems are solved by an algorithm based on the theoretical framework of the augmented Lagrangian SQP algorithm given by Murray and Prieto in "A second-derivative method for nonlinearly constrained optimization" (SOL Technical Report 95-3). The absence of inequality constraints allows much more flexibility in how the search directions may be computed, which is paramount when solving large problems. This framework guarantees convergence to a second-order point when a direction of negative curvature for the reduced Hessian is used in conjunction with a descent direction. A null-space method is used to obtained the search directions. Specifically the conjugate gradient algorithm is used to compute a descent direction, to detect indefiniteness in the reduced Hessian and to compute a direction of negative curvature. This direction of negative curvature is improved by application of a few steps of the Lanczos algorithm. A linesearch is then performed along the curve formed by a combination of the descent direction and the direction of negative curvature to get a steplength. A key feature of the algorithm is the scaling of each subproblem. It enhances the performance of the conjugate gradient algorithm and has other benefits. IPSOL has been prototyped in Matlab and tested on a subset of the CUTEr test problems. This thesis is focused on the algorithm and the details of implementation of IPSOL. We also discuss its performance on the CUTEr test set and compare the results against the current generation barrier solvers LOQO and IPOPT.