Robust optimization and stochastic optimization are the two main paradigms for dealing with the uncertainty inherent in almost all real-world optimization problems. The core principle of robust optimization is the introduction of parameterized families of constraints. Sometimes, these complicated semi-infinite constraints can be reduced to finitely many convex constraints, so that the resulting optimization problem can be solved using standard procedures. Hence flexibility of robust optimization is limited by certain convexity requirements on various objects. However, a recent strain of literature has sought to expand applicability of robust optimization by lifting variables to a properly chosen matrix space. Doing so allows to handle situations where convexity requirements are not met immediately, but rather intermediately.

In the domain of (possibly nonconvex) quadratic optimization, the principles of copositive optimization act as a bridge leading to recovery of the desired convex structures. Copositive optimization has established itself as a powerful paradigm for tackling a wide range of quadratically constrained quadratic optimization problems, reformulating them into linear convex-conic optimization problems involving only linear constraints and objective, plus constraints forcing membership to some matrix cones, which can be thought of as generalizations of the positive-semidefinite matrix cone. These reformulations enable application of powerful optimization techniques, most notably convex duality, to problems which, in their original form, are highly nonconvex.

In this text we want to offer readers an introduction and tutorial on these principles of copositive optimization, and to provide a review and outlook of the literature that applies these to optimization problems involving uncertainty.

鲁棒优化和随机优化是处理几乎所有现实世界优化问题中固有的不确定性的两个主要范例。稳健优化的核心原则是引入参数化约束族。有时，这些复杂的半无限约束可以简化为有限多个凸约束，因此可以使用标准程序解决生成的优化问题。因此，鲁棒优化的灵活性受到各种对象的某些凸性要求的限制。然而，最近的一系列文献试图通过将变量提升到适当选择的矩阵空间来扩展稳健优化的适用性。这样做可以处理凸性要求不是立即满足而是中间满足的情况。

在（可能是非凸的）二次优化领域，余优化的原理充当了导致恢复所需凸结构的桥梁。共正优化已成为解决广泛的二次约束二次优化问题的强大范例，将它们重新表述为仅涉及线性约束和目标的线性凸圆锥优化问题，加上强制隶属于某些矩阵锥的约束，可以认为作为半正定矩阵锥的推广。这些重新表述使强大的优化技术（最著名的是凸对偶性）能够应用于原始形式高度非凸的问题。

在本文中，我们希望为读者提供有关余正优化原则的介绍和教程，并对将这些原则应用于涉及不确定性的优化问题的文献进行回顾和展望。