二次规划 Quadratic Programming,二次规划,Questions,什么是二次规划？ 如何求解？,二次规划,Definition,二次规划,（*）,二次规划,Remark,二次规划,等式约束二次规划问题的解法： 直接消去法(Direct Elimination Method) Lagrange乘子法,直接消去法(Direct Elimination Method),直接消去法,直接消去法,直接消去法,(*),直接消去法,直接消去法,直接消去法,Remark,直接消去法,Example,直接消去法,Solution,直接消去法,直接消去法,直接消去法,Remark,Lagrange乘子法,Lagrange乘子法,Lagrange乘子法,Lagrange乘子法,Lagrange乘子法,把c和b代入前面的最优解然后利用H R即可得。,Lagrange乘子法,Example,Lagrange乘子法,Solution,Lagrange乘子法,Lagrange乘子法,起作用集方法（Active Set Method),起作用集方法,正定二次规划,（*）,定 理,起作用集方法,Theorem,起作用集方法,Theorem,(*),起作用集方法,Proof,起作用集方法,（a）,起作用集方法,(b),满足（a)的 肯定满足（b），且为满足（b）的 的一部分，但满足（b）的解是唯一的，所以问题（b）的解就是问题（a)的解。,起作用集方法,Remark,起作用集方法,Questions,起作用集方法,起作用集方法,(*),起作用集方法,起作用集方法,(*),起作用集方法,起作用集方法,起作用集方法,起作用集方法,Questions,如何得到(*)？,起作用集方法,Answer,起作用集方法,Questions,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,Algorithm,Algorithm,Algorithm,Algorithm,起作用集方法,起作用集方法,起作用集方法,起作用集方法,Example,起作用集方法,Solution,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,起作用集方法,序列二次规划法 （Sequential Quadratic Programming),（约束拟牛顿法约束变尺度法）,序列二次规划法,序列二次规划法,序列二次规划法,序列二次规划法,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,牛顿法求方程组 的解,序列二次规划法,序列二次规划法,序列二次规划法,Remark,序列二次规划法,Remark,序列二次规划法,Example,序列二次规划法,序列二次规划法,序列二次规划法,序列二次规划法,序列二次规划法,用信赖域方法求解二次规划子问题：,序列二次规划法,序列二次规划法,序列二次规划法,(*),序列二次规划法,序列二次规划法,序列二次规划法,序列二次规划法,序列二次规划法,序列二次规划法,(1),General Problem (GP),序列二次规划法,In constrained optimization, the general aim is to transform the problem into an easier sub-problem that can then be solved and used as the basis of an iterative process.,序列二次规划法,A characteristic of a large class of early methods is the translation of the constrained problem to a basic unconstrained problem by using a penalty function for constraints that are near or beyond the constraint boundary. In this way the constrained problem is solved using a sequence of parameterized unconstrained optimizations, which in the limit (of the sequence) converge to the constrained problem.,序列二次规划法,These methods are now considered relatively inefficient and have been replaced by methods that have focused on the solution of the Kuhn-Tucker (KT) equations. The KT equations are necessary conditions for optimality for a constrained optimization problem.,序列二次规划法,If the problem is a so-called convex programming problem, then the KT equations are both necessary and sufficient for a global solution point.,序列二次规划法,Referring to GP (1), the Kuhn-Tucker equations can be stated as,(2),序列二次规划法,The first equation describes a canceling of the gradients between the objective function and the active constraints at the solution point. For the gradients to be canceled, Lagrange multipliers are necessary to balance the deviations in magnitude of the objective function and constraint gradients.,序列二次规划法,Because only active constraints are included in this canceling operation, constraints that are not active must not be included in this operation and so are given Lagrange multipliers equal to zero. This is stated implicitly in the last two equations of Eq. (2).,序列二次规划法,The solution of the KT equations forms the Basis to many nonlinear programming algorithms. These algorithms attempt to compute the Lagrange multipliers directly. Constrained quasi-Newton methods guarantee super-linear convergence by accumulating second order information regarding the KT equations using a quasi-Newton updating procedure.,序列二次规划法,These methods are commonly referred to as Sequential Quadratic Programming (SQP) methods, since a QP sub-problem is solved at each major iteration (also known as Iterative Quadratic Programming, Recursive Quadratic Programming, and Constrained Variable Metric methods).,序列二次规划法,Sequential Quadratic Programming (SQP) A Quadratic Programming (QP) Subproblem SQP Implementation,序列二次规划法,Sequential Quadratic Programming (SQP) A Quadratic Programming (QP) Subproblem SQP Implementation,序列二次规划法,Sequential Quadratic Programming (SQP)SQP methods represent the state of the art in nonlinear programming methods. Schittkowski , for example, has implemented and tested a version that outperforms every other tested method in terms of efficiency, accuracy, and percentage of successful solutions, over a large number of test problems.,序列二次规划法,Based on the work of Biggs , Han, and Powell , the method allows you to closely mimic Newtons method for constrained optimization just as is done for unconstrained optimization. At each major iteration, an approximation is made of the Hessian of the Lagrangian function using a quasi-Newton updating method.,序列二次规划法,This is then used to generate a QP sub-problem whose solution is used to form a search direction for a line search procedure. The general method, however, is stated here.,序列二次规划法,Given the problem description in GP (1) the principal idea is the form