Lineárisquadratikus

Lineárisquadratikus is a term used to describe models, equations or optimization problems that combine linear and quadratic components. In mathematical contexts it often refers to functions or objectives that include both linear terms and a quadratic term, capturing first-order effects as well as curvature or interactions.

A common mathematical form is f(x) = c^T x + (1/2) x^T Q x + r, where x is a vector in R^n, c is a coefficient vector, Q is a symmetric matrix, and r is a scalar constant. The linear part c^T x represents direct, proportional effects, while the quadratic part (1/2) x^T Q x accounts for interactions between variables and the function’s curvature. The properties of the quadratic term largely determine convexity: if Q is positive semidefinite, f is convex; if Q is positive definite, f is strictly convex; if Q has negative eigenvalues, the function may be nonconvex.

In optimization, lineárisquadratikus problems typically involve minimizing such an objective under linear constraints, for example Ax ≤ b or Ax = b. These problems are widely categorized as quadratic programming (QP). For convex QP (Q positive semidefinite), efficient global-solvers exist; for nonconvex cases, one may obtain local minima and rely on specialized algorithms or global optimization techniques.

Applications of lineárisquadratikus models appear in control theory (for instance, linear-quadratic regulator problems), statistics and machine learning (regularized or ridge-type objectives), economics, and finance, where both linear effects and quadratic interactions are relevant. The concept is a foundational building block in convex and nonconvex optimization frameworks.