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- Cooperative optimization is a global optimization method invented by Chinese mathematician Xiao-Fei Huang, that can solve real-world NP-hard optimization problems (up to millions of variables) with outstanding performances and unprecedented speeds. It knows whether a solution it found is a global optimum and which direction is more promising than others for finding a global optimum. With some very general settings, a cooperative optimization algorithm has a unique equilibrium and converges to it regardless of initial conditions and is insensitive to perturbations. It does not struggle with local minima. Cooperative optimization is based on a system of multiple agents cooperating to solve a hard optimization problem. The problem is divided into a number of manageable sub-problems which are each assigned to an agent. The cooperation is achieved by asking the agents to pass their solutions to their neighbors and to compromise their solutions if they conflict. When all conflicts are resolved, the system has found a consensus solution as the solution to the original problem. For a certain class of cooperative optimization algorithms, a consensus solution must be the global optimal one, guaranteed by theory. The process of a cooperative system of this kind is iterative and self-organized and each agent in the system is autonomous. The system is also inherently distributed and parallel, making the entire system highly scalable and less vulnerable to perturbations and disruptions than a centralized system.
- 协同优化算法的原理是将一复杂的目标函数分解成简单的子目标函数，然后再将这些子目标函数进行协同优化。具体说来，协同优化是在优化每一子目标函数同时综合考虑其它子目标函数的结果，使子目标函数之间的优化结果能够一致。优化结果一致是指使每一变量的值在每一子目标函数的优化结果中能够一致。一般来说，可以证明，如果变量的值一致则为最优解。协同优化算法没有局部最优问题同时具有非常良好的收敛特性。 它很好地解决了许多实际中非线性优化及组合优化难题。 如果目标函数是一n个变量的函数<math>E(x_1, x_2, \ldots, x_n)</math>,简写成<math>E(x)</math>,协同优化算法先将它分解成n个简单的子目标函数: <math>E(x) = E_1(x) + E_2(x) + \ldots + E_n(x)</math>. 如果单独优化每一子目标函数,则它们的结果很难达到一致。例如,变量<math>x_i</math>在包含它的子目标函数中的最优解值很难相同。对于<math>i=1,2,\ldots,n</math>,如果我们取<math>E_i(x)</math>的最优解中<math>x_i</math>的值作为该变量的值，表示成<math>\tilde{x}_i</math>, <math>\tilde{x}_i = \arg \min_{x_i} \min_{X_i \setminus x_i} E_i(x)</math>，这里，<math>X_i</math>是<math>E_i(x)</math>的变量集，<math>X_i \setminus x_i</math>指变量集<math>X_i</math>除去元素<math>x_i</math>， <math>(\tilde{x}_1,\tilde{x}_2,\ldots,\tilde{x}_n)</math>则很难为原目标函数<math>E(x)</math>的最优解。 为了使子目标函数之间的优化结果能够一致，协同优化算法在优化每一子目标函数<math>E_i(x)</math>同时考虑其它子目标函数的结果, <math>\Psi_j(x_j) = \min_{X_j \setminus x_j} E_j(x)</math>。 具体做法是利用其它子目标函数的优化结果通过数值加权修正每一个子目标函数如下, <math>\left(1 - \lambda \right) E_i(x) + \lambda_k \sum_{j} w_{ij} \Psi_j(x_j)</math>，这里，<math>\lambda_k,w_{ij}</math>为加权系数，满足<math>0 \le \lambda_k,w_{ij} \le 1</math>。 然后对修正后的子目标函数进行优化，优化结果再叠代放入修正的子目标函数中。协同优化算法的叠代方程如下, <math>\Psi^{(k)}_i (x_i) = \min_{X_i \setminus{x_i}}\left(\left E_i +\lambda_k \sum_{j} w_{ij} \Psi^{}_j\right), \quad for \; i=1,2,\ldots,n. </math> 协同优化结果使每一变量的值在每一子目标函数的优化结果中达到一致。如果一致，则子目标函数的优化解既为最优解。
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