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Convex Quadratic Objective Function

A convex quadratic objective function is defined by the general mathematical form $h(\mathbf{x}) = \frac{1}{2} \mathbf{x}^ op \mathbf{Q} \mathbf{x} + \mathbf{x}^ op \mathbf{c} + b$ , where the matrix $\mathbf{Q}$ is positive definite ( $\mathbf{Q} \succ 0$ ). Because $\mathbf{Q}$ possesses strictly positive eigenvalues, this function has a unique global minimizer located at $\mathbf{x}^* = -\mathbf{Q}^{-1} \mathbf{c}$ . The function can be rewritten by centering it around this minimizer, yielding $h(\mathbf{x}) = \frac{1}{2} (\mathbf{x} - \mathbf{Q}^{-1} \mathbf{c})^ op \mathbf{Q} (\mathbf{x} - \mathbf{Q}^{-1} \mathbf{c}) + b - \frac{1}{2} \mathbf{c}^ op \mathbf{Q}^{-1} \mathbf{c}$ . Furthermore, its gradient is given by $\partial_{\mathbf{x}} h(\mathbf{x}) = \mathbf{Q} (\mathbf{x} - \mathbf{Q}^{-1} \mathbf{c})$ , which geometrically represents the distance from the point $\mathbf{x}$ to the minimizer scaled by the curvature matrix $\mathbf{Q}$ .

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Updated 2026-05-15

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