Canonical Duality Theory for Global Optimization problems and applications

The canonical duality theory is studied, through a discussion on a general global optimization problem and applications on fundamentally important problems. This general problem is a formulation of the minimization problem with inequality constraints, where the objective function and constraints are any convex or nonconvex functions satisfying certain decomposition conditions. It covers convex problems, mixed integer programming problems and many other nonlinear programming problems. The three main parts of the canonical duality theory are canonical dual transformation, complementary-dual principle and triality theory. The complementary-dual principle is further developed, which conventionally states that each critical point of the canonical dual problem is corresponding to a KKT point of the primal problem with their sharing the same function value. The new result emphasizes that there exists a one-to-one correspondence between KKT points of the dual problem and of the primal problem and each pair of the corresponding KKT points share the same function value, which implies that there is truly no duality gap between the canonical dual problem and the primal problem. The triality theory reveals insightful information about global and local solutions. It is shown that as long as the global optimality condition holds true, the primal problem is equivalent to a convex problem in the dual space, which can be solved efficiently by existing convex methods; even if the condition does not hold, the convex problem still provides a lower bound that is at least as good as that by the Lagrangian relaxation method. It is also shown that through examining the canonical dual problem, the hidden convexity of the primal problem is easily observable. The canonical duality theory is then applied to dealing with three fundamentally important problems. The first one is the spherically constrained quadratic problem, also referred to as the trust region subproblem. The canonical dual problem is onedimensional and it is proved that the primal problem, no matter with convex or nonconvex objective function, is equivalent to a convex problem in the dual space. Moreover, conditions are found which comprise the boundary that separates instances into “hard case” and “easy case”. A canonical primal-dual algorithm is developed, which is able to efficiently solve the problem, including the “hard case”, and can be used as a unified method for similar problems. The second one is the binary quadratic problem, a fundamental problem in discrete optimization. The discussion is focused on lower bounds and analytically solvable cases, which are obtained by analyzing the canonical dual problem with perturbation techniques. The third one is a general nonconvex problem with log-sum-exp functions and quartic polynomials. It arises widely in engineering science and it can be used to approximate nonsmooth optimization problems. The work shows that problems can still be efficiently solved, via the canonical duality approach, even if they are nonconvex and nonsmooth.Doctor of Philosoph

An enumerative method for convex programs with linear complementarity constraints and application to the bilevel problem of a forecast model for high complexity products

The increasing variety of high complexity products presents a challenge in acquiring detailed demand forecasts. Against this backdrop, a convex quadratic parameter dependent forecast model is revisited, which calculates a prognosis for structural parts based on historical order data. The parameter dependency inspires a bilevel problem with convex objective function, that allows for the calculation of optimal parameter settings in the forecast model. The bilevel problem can be formulated as a mathematical problem with equilibrium constraints (MPEC), which has a convex objective function and linear constraints. Several new enumerative methods are presented, that find stationary points or global optima for this problem class. An algorithmic concept shows a recursive pattern, which finds global optima of a convex objective function on a general non-convex set defined by a union of polytopes. Inspired by these concepts the thesis investigates two implementations for MPECs, a search algorithm and a hybrid algorithm

Numerical Methods for Mixed-Integer Optimal Control with Combinatorial Constraints

This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing- Constraint respecting rounding algorithm to exploit this correspondence computationally. Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions. For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gauß-Newton preconditioner for the iterative solution method of the trust region problem. We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems