半代数系统的求解是实代数几何研究的主要对象之一，它是符号计算领域最活跃的研究课题之一。计算有界全维数的半代数系统的全部整数解集在非线性程序优化和程序自动并行化方面具有重要的应用价值。目前求解半代数系统全部解的方法几乎是符号方法，符号方法具有计算结果准确的优势，但不足之处就是计算复杂度高，在实际应用中效率低，只能求解小规模的半代数系统。本项目将采用全数值的方法研究有界的全维数的半代数系统全部整数解问题，通过本项目的实施，我们将初步建立起半代数集数值投影的理论基础，发展零误差计算的高效率算法，设计出高效的多重循环的模型来描述和遍历全部整数解的方法。本项目提出的方法可避免符号算法的中间表达式膨胀问题，易于在目前流行的编程语言上实现，为非线性程序优化、自动并行化等领域提供理论支持和计算工具。

Solving semi-algebraic systems is one of the main research objects in real algebraic geometry. It is one of the most active research topics in symbolic computation. Finding all the integer solutions of a bounded full-dimensional semi-algebraic system has important applications in the fields of nonlinear program optimization, automatic parallelization, etc. Most of the current methods for such problems are symbolic, which can produce the exact results. However, the high complexity of the symbolic methods limits their applications to small size problems. Instead, in this project we will make use of pure numerical methods to study such problems. This project will help initializing the numerical projection theory of semi-algebraic sets, developing efficient algorithms for zero-error computation, and designing an efficient multi-nested loop model to describe and enumerate all the integer solutions. The proposed idea in this project can avoid the expression swell problem occurring in symbolic computation. It is also easy to be implemented in mainstream programming languages. This project will provide theoretical support and computational tools for nonlinear program optimization, automatic parallelization and other applications.