Convex Quadratic Equation
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Convex Quadratic Equation Li-Gang Lin1
· Yew-Wen Liang2 · Wen-Yuan Hsieh3
Received: 17 February 2020 / Accepted: 19 July 2020 / Published online: 27 August 2020 © The Author(s) 2020
Abstract Two main results (A) and (B) are presented in algebraic closed forms. (A) Regarding the convex quadratic equation, an analytical equivalent solvability condition and parameterization of all solutions are formulated, for the first time in the literature and in a unified framework. The philosophy is based on the matrix algebra, while facilitated by a novel equivalence/coordinate transformation (with respect to the much more challenging case of rank-deficient Hessian matrix). In addition, the parameter-solution bijection is verified. From the perspective via (A), a major application is re-examined that accounts for the other main result (B), which deals with both the infinite and finite-time horizon nonlinear optimal control. By virtue of (A), the underlying convex quadratic equations associated with the Hamilton–Jacobi equation, Hamilton–Jacobi inequality, and Hamilton–Jacobi–Bellman equation are explicitly solved, respectively. Therefore, the long quest for the constituent of the optimal controller, gradient of the associated value function, can be captured in each solution set. Moving forward, a preliminary to exactly locate the optimality using the state-dependent (resp., differential) Riccati equation scheme is prepared for the remaining symmetry condition. Keywords Convex quadratic equation · Matrix algebra · Optimal control · Nonlinear system · Convex quadratic function Mathematics Subject Classification 15A18 · 49J20 · 49N35 · 52A41 · 93C10 · 93C35
Communicated by Kok Lay Teo.
B
Li-Gang Lin [email protected]; [email protected] Yew-Wen Liang [email protected] Wen-Yuan Hsieh [email protected]
1
Department of Mechanical Engineering, National Central University, Taoyuan, Taiwan
2
Institute of Electrical Control Engineering, National Chiao Tung University, Hsinchu, Taiwan
3
Sonix Technology Co., Ltd, Zhubei, Taiwan
123
Journal of Optimization Theory and Applications (2020) 186:1006–1028
1007
1 Introduction As a mathematical fundamental, the convex quadratic function (CQF) appears in a variety of topics and applications [1]. For example, in the field of matrix analysis, two basic properties “positive definiteness and semidefiniteness” are within the scope. In addition, regarding the field of optimization, if the objective function is convex and quadratic, then it falls into the categorization of nonlinear programming or, more fundamentally, the quadratic programming (QP) [2], which includes the linear programming as a special case. More generally, if a convex function is sufficiently differentiable, then its local behavior can resemble a quadratic one [3], which benefits existing optimization algorithms. Notably, subject to equality and/or inequality constraints, the QP constitutes the basis for an extension of the renowned Newton’s method [4]. Associated with the CQF, the convex quadratic equation (CQE)
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