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We propose an algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables. Each iteration of the algorithm involves solving a linear program with complementarity constraints in order to obtain an estimate of the active set. The algorithm enforces descent on the objective function to promote global convergence to B-stationary points. We provide a convergence analysis and preliminary numerical results on a range of test problems. We also study the effect of fixing the active constraints in a bound-constrained quadratic program that can be solved on each iteration in order to obtain fast convergence. [ABSTRACT FROM AUTHOR]

We consider the use of the polynomial chaos method to approximate optimal control problems with randomly varying uncertain parameters by deterministic surrogate problems. Our focus is on nonlinear problems that require a high expansion order to give meaningful statements about the uncertainty propagation. Their resulting size and complexity pose a computational challenge for traditional optimal control methods. Our contributions include an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision, as is highlighted in a numerical case study with two nonlinear real-world problems. The algorithmic contribution is complemented by a convergence proof showing that the optimal control solutions approach the correct solution for increasing expansion orders. [ABSTRACT FROM AUTHOR]

Partial outer convexification has been used to derive relaxations of mixed-integer optimal control problems (MIOCPs) that are constrained by time-dependent differential equations. The family of sum-up rounding (SUR) algorithms provides a means to approximate feasible points of these relaxations, i.e., [0,1]-valued control trajectories, with {0,1}-valued points. The approximants computed by an SUR algorithm converge in a weak sense when the coarseness of the rounding grid of the SUR algorithm is driven to zero, which in turn induces norm convergence of the corresponding sequence of state vectors. We show that this approximation property can be transferred to MIOCPs with integer control variables distributed in more than one dimension when carrying out an appropriate grid refinement strategy. We deduce a norm convergence result for the state vector of elliptic PDE systems and provide computational results illustrating the applicability of the theoretical framework. [ABSTRACT FROM AUTHOR]

A new method for solving sequences of quadratic programs (QPs) is presented. For each new QP in the sequence, the method utilizes hot-starts that employ information computed by an active-set QP solver during the solution of the first QP. This avoids the computation and factorization of the full constraint and Hessian matrices for all but the first problem in the sequence. The proposed algorithm can be seen as an extension of the iterative refinement procedure for linear systems to QP problems, coupled with the application of an accelerated linear solver method that employs hot-started QP solves as a preconditioner. Local convergence results are presented. The practical performance of the proposed method is demonstrated on a sequence of QPs arising in nonlinear model predictive control and during the solution of a set of randomly generated nonlinear optimization problems using sequential quadratic programming. In these experiments, the method proves to be fairly reliable, despite the lack of global convergence guarantees. The results also show a significant reduction in the computation time for large problems with dense constraint matrices, as well as in the number of matrix-vector products. [ABSTRACT FROM AUTHOR]