Convex Quadratic Equations and Functions

Abstract

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