A framework is proposed for solving general convex quadratic programs (CQPs) from an infeasible starting point by invoking an existing feasible-start algorithm tailored for inequality-constrained CQPs. The central tool is an exact penalty function scheme equipped with a penalty-parameter updating rule. The feasible-start algorithm merely has to satisfy certain general requirements, and so is the updating rule. Under mild assumptions, the framework is proved to converge on CQPs with both inequality and equality constraints and, at a negligible additional cost per iteration, produces an infeasibility certificate, together with a feasible point for an (approximately) \(\ell _1\)-least relaxed feasible problem, when the given problem does not have a feasible solution. The framework is applied to a feasible-start constraint-reduced interior-point algorithm previously proved to be highly performant on problems with many more inequality constraints than variables (“imbalanced”). Numerical comparison with popular codes (OSQP, qpOASES, MOSEK) is reported on both randomly generated problems and support-vector machine classifier training problems. The results show that the former typically outperforms the latter on imbalanced problems. Finally, application of the proposed infeasible-start framework to other feasible-start algorithms is briefly considered, and is tested on a simplex iteration.

提出了一种框架，用于通过调用为不等式约束 CQP量身定制的现有可行开始算法，从不可行的起点解决一般凸二次规划 (CQP) 。中心工具是配备了惩罚参数更新规则的精确惩罚函数方案。可行启动算法只需要满足某些一般要求，更新规则也是如此。在温和的假设下，该框架被证明在具有不等式和等式约束的 CQP 上收敛，并且在每次迭代的额外成本可以忽略不计的情况下，产生不可行性证书，以及（大约）\(\ell _1\)- 最小松弛可行问题，当给定的问题没有可行解时。该框架应用于可行起始约束减少内点算法，先前已证明它在具有比变量更多的不等式约束（“不平衡”）的问题上具有很高的性能。在随机生成的问题和支持向量机分类器训练问题上都报告了与流行代码（OSQP、qpOASES、MOSEK）的数值比较。结果表明，前者在不平衡问题上通常优于后者。最后，简要考虑了所提出的不可行启动框架在其他可行启动算法中的应用，并在单纯形迭代上进行了测试。