Gibonacci Optimization : duality (Mathematical Decision Making Under Uncertainty and Related Topics)

We show that a parametric linear system of equations plays a fundamental part in establishing a mutual relation between minimization problem (primal) and maximization problem (dual). The system is of 2n-equation on 2n-variable, called zero-minimum condition. It yields a couple of second-order finite (n-) linear difference equation on n-variable, which constitute the respective optimal conditions. The respective equations have a mimimum solution for primal and a maximum one for dual. Both the optimal solutions are expressed in terms of Gibonacci sequence, which is a parametric generalization of the Fibonacci one. Either solution is characterized by the backward Gibonacci sequence and its complementary --Hibonacci sequence--

WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction

WavePacket is an open-source program package for numeric simulations in quantum dynamics. It can solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows, e.g., to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semi-classical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. Being highly versatile and offering visualization of quantum dynamics 'on the fly', WavePacket is well suited for teaching or research projects in atomic, molecular and optical physics as well as in physical or theoretical chemistry. Building on the previous Part I which dealt with closed quantum systems and discrete variable representations, the present Part II focuses on the dynamics of open quantum systems, with Lindblad operators modeling dissipation and dephasing. This part also describes the WavePacket function for optimal control of quantum dynamics, building on rapid monotonically convergent iteration methods. Furthermore, two different approaches to dimension reduction implemented in WavePacket are documented here. In the first one, a balancing transformation based on the concepts of controllability and observability Gramians is used to identify states that are neither well controllable nor well observable. Those states are either truncated or averaged out. In the other approach, the H2-error for a given reduced dimensionality is minimized by H2 optimal model reduction techniques, utilizing a bilinear iterative rational Krylov algorithm

Triplet of Fibonacci Duals : with or without constraint (New Developments on Mathematical Decision Making Under Uncertainty)

We consider a dual relation between minimization (primal) problem and maximization (dual) problem from a view point of complementarity. An identity (CI) [n-1]Σ[k=1][(xk-1 - xk)μk + xk(μk - μk+1)] + (xn-1 - xn)μn + xnμn = x0μ1 is called complementary [20, 22]. We present three types of complementary identities, which take a fundamental role in analyzing respective pairs of primal and dual. Moreover, we show that a primal and its dual satisfy Fibonacci Complementary Duality [18, 19, 21, 22]

Hidden Convexity in Partially Separable Optimization

The paper identifies classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some partial separable structure. Among other things, the new hidden convexity results open up the possibility to solve multi-stage robust optimization problems using certain nonlinear decision rules.convex relaxation of nonconvex problems;hidden convexity;partially separable functions;robust optimization