Solvers for Separable and Equality QP/QCQP Problems

• Published in: Scalable Algorithms for Contact Problems Vol. 36; pp. 135 - 160
• Main Authors: Sadowská, Marie, Dostál, Zdeněk, Vondrák, Vít, Kozubek, Tomás
• Format: Book Chapter
• Language: English
• Published: United States Springer New York 2017
• Series: Advances in Mechanics and Mathematics
• Subjects: Abadie Constraint Qualification; Active Bound Constraints; Augmented Lagrangian; Feasible Error; Infinite series; Mathematical theory of computation; Maths for engineers; Modified Cost Function
• ISBN: 9781493968329; 1493968327
• ISSN: 1571-8689; 1876-9896
• DOI: 10.1007/978-1-4939-6834-3_9

We shall now use the results of our previous investigations to develop efficient algorithms for the minimization of strictly convex quadratic functions subject to possibly nonlinear convex separable inequality constraints and linear equality constraints $$\min \limits _{\mathbf {x}\in \varOmega _{SE}} f(\mathbf {x}), \,\, f(\mathbf {x})=\frac{1}{2} \mathbf {x}^T\mathsf A\mathbf x-\mathbf x^T\mathbf b,$$ where $$\varOmega _{SE}=\{\mathbf {x}\in \mathbb {R}^n: \mathsf {B}\mathbf {x}=\mathbf {o} \ \ \mathrm {and} \ \ \mathbf {x}\in \varOmega _S\}, \,\, \varOmega _S=\{\mathbf {x}\in \mathbb {R}^n: h_i(\mathbf {x}_i)\le 0, \ i=1, \dots ,s\},$$   $$\mathbf b\in \mathbb {R}^n$$ ,  $$h_i$$ are convex functions, $$\mathsf A$$ is an $$n\times n$$ SPD matrix, and $$\mathsf {B}\in \mathbb {R}^{m\times n}$$ .