3 research outputs found
Convex Quadratic Equations and Functions
Three interconnected main results (1)-(3) are presented in closed forms. (1)
Regarding the convex quadratic equation (CQE), an analytical equivalent
solvability condition and parameterization of all solutions are completely
formulated, in a unified framework. The design concept is based on the matrix
algebra, while facilitated by a novel equivalence/coordinate transformation.
Notably, the parameter-solution bijection is also verified. Two applications
are selected as the other two main results. (2) The focus is set on both the
infinite and finite-time horizon nonlinear optimal control. By virtue of (1),
the CQEs associated with the underlying Hamilton-Jacobi Equation,
Hamilton-Jacobi Inequality, and Hamilton-Jacobi-Bellman Equation are
algebraically solved, respectively. Each solution set captures the gradient of
the associated value function. Moving forward, a preliminary to recover the
optimality using the state-dependent (differential) Riccati equation is
provided, which can also be used to more efficiently implement the last main
result. (3) The nonlinear programming/convex optimization is analyzed via a
novel method and perspective. The philosophy is based on the new analysis of
CQE in (1), which helps explain the geometric structure of the convex quadratic
function (CQF), and the CQE-CQF relation. An impact on the quadratic
programming (QP), a basis in the literature, is demonstrated. Specifically, the
QPs subject to equality, inequality, equality-and-inequality, and extended
constraints are algebraically solved, resp., without knowing a feasible point.Comment: This manuscript is only preliminary and still growing. Therefore,
with expectations, we deeply appreciate all kinds of input