\lim \limits _{(x,y) \in R^{2}}f(x)=y-(x-2)^{2}+3 s. t. y ≥ 1 计算满足KKT条件的点,并利用二阶条件验证上述点是否是局部极小值点
时间: 2024-05-28 08:12:56 浏览: 14
根据KKT条件,可得以下条件:
1. 梯度为0:$\nabla f(x,y)=\begin{bmatrix} -2(x-2) \\ 1 \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \end{bmatrix}$
2. $y-1 \geq 0$
3. $\lambda(y-1)=0$
4. $\lambda \geq 0$
解以上方程组,可得:
当 $x=2$,$y=3$ 时,$\nabla f(x,y)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$,满足条件2和3,$\lambda=0$,满足条件4。因此,$(2,3)$是满足KKT条件的点。
对$f(x,y)$求二阶偏导数,得:
$\dfrac{\partial^{2}f}{\partial x^{2}}=-2$
$\dfrac{\partial^{2}f}{\partial y^{2}}=0$
$\dfrac{\partial^{2}f}{\partial x \partial y}=0$
因此,Hessian矩阵为$\begin{bmatrix} -2 & 0 \\ 0 & 0 \end{bmatrix}$,而且$\dfrac{\partial^{2}f}{\partial x^{2}}<0$,因此$(2,3)$是$f(x,y)$的一个局部极小值点。
相关问题
\lim \limits _{(x,y) \in IR^{2}}f(x)=(x-1)^{2}+y-2 s. t. h( x) = y - x - 1 = 0 g(x) =x+y - 2≤ 0. 计算满足KKT条件的点,并利用二阶条件验证上述点是否是局部极小值点.
首先列出拉格朗日函数:
$$
L(x,y,\alpha,\beta)=f(x)-\alpha h(x)-\beta g(x)
$$
其中,$\alpha$ 和 $\beta$ 是拉格朗日乘子。
根据KKT条件,有:
$$
\begin{aligned}
&\nabla f(x)-\alpha \nabla h(x)-\beta \nabla g(x)=0\\
&h(x)=0\\
&g(x)\le 0\\
&\beta\ge 0\\
&\beta g(x)=0\\
\end{aligned}
$$
将$f(x)$和$h(x)$代入,可得:
$$
\begin{aligned}
&\begin{cases}
2(x-1)-\alpha-\beta=0\\
1-\alpha=0\\
x+y-2\le 0\\
\beta\ge 0\\
\beta(x+y-2)=0\\
\end{cases}\\
\Rightarrow&\begin{cases}
\alpha=1\\
\beta=0\\
x+y-2\le 0\\
\end{cases}
\end{aligned}
$$
因此,满足KKT条件的点为:
$$
(x,y)=(1,1)
$$
接下来,利用二阶条件验证$(1,1)$是否是局部极小值点。
首先,计算$\nabla^2 f(x)$:
$$
\nabla^2 f(x)=\begin{pmatrix}
2 & 0 \\
0 & 1 \\
\end{pmatrix}
$$
然后,计算$\nabla h(x)$和$\nabla g(x)$:
$$
\nabla h(x)=\begin{pmatrix}
-1 \\
1 \\
\end{pmatrix}
$$
$$
\nabla g(x)=\begin{pmatrix}
1 \\
1 \\
\end{pmatrix}
$$
接着,计算$\nabla^2 L(x)$:
$$
\nabla^2 L(x)=\nabla^2 f(x)-\alpha \nabla^2 h(x)-\beta \nabla^2 g(x)
$$
$$
\nabla^2 L(x)=\begin{pmatrix}
2 & 0 \\
0 & 1 \\
\end{pmatrix}-\begin{pmatrix}
0 & 0 \\
0 & 0 \\
\end{pmatrix}-\begin{pmatrix}
0 & 0 \\
0 & 0 \\
\end{pmatrix}=\begin{pmatrix}
2 & 0 \\
0 & 1 \\
\end{pmatrix}
$$
因为$\nabla^2 L(x)$是正定的,所以$(1,1)$是局部极小值点。
给下列程序添加注释: void DWAPlannerROS::reconfigureCB(DWAPlannerConfig &config, uint32_t level) { if (setup_ && config.restore_defaults) { config = default_config_; config.restore_defaults = false; } if ( ! setup_) { default_config_ = config; setup_ = true; } // update generic local planner params base_local_planner::LocalPlannerLimits limits; limits.max_vel_trans = config.max_vel_trans; limits.min_vel_trans = config.min_vel_trans; limits.max_vel_x = config.max_vel_x; limits.min_vel_x = config.min_vel_x; limits.max_vel_y = config.max_vel_y; limits.min_vel_y = config.min_vel_y; limits.max_vel_theta = config.max_vel_theta; limits.min_vel_theta = config.min_vel_theta; limits.acc_lim_x = config.acc_lim_x; limits.acc_lim_y = config.acc_lim_y; limits.acc_lim_theta = config.acc_lim_theta; limits.acc_lim_trans = config.acc_lim_trans; limits.xy_goal_tolerance = config.xy_goal_tolerance; limits.yaw_goal_tolerance = config.yaw_goal_tolerance; limits.prune_plan = config.prune_plan; limits.trans_stopped_vel = config.trans_stopped_vel; limits.theta_stopped_vel = config.theta_stopped_vel; planner_util_.reconfigureCB(limits, config.restore_defaults); // update dwa specific configuration dp_->reconfigure(config); }
/**
* @brief Callback function for dynamic reconfiguration of DWA planner parameters
*
* @param config Reference to the configuration object that stores the updated parameters
* @param level The level of reconfiguration, unused in this function
*/
void DWAPlannerROS::reconfigureCB(DWAPlannerConfig &config, uint32_t level) {
// If the setup has been completed and restore_defaults flag is set, restore default configuration
if (setup_ && config.restore_defaults) {
config = default_config_;
config.restore_defaults = false;
}
// If setup has not been completed, store default configuration and set the setup flag to true
if ( ! setup_) {
default_config_ = config;
setup_ = true;
}
// Update generic local planner parameters
base_local_planner::LocalPlannerLimits limits;
limits.max_vel_trans = config.max_vel_trans;
limits.min_vel_trans = config.min_vel_trans;
limits.max_vel_x = config.max_vel_x;
limits.min_vel_x = config.min_vel_x;
limits.max_vel_y = config.max_vel_y;
limits.min_vel_y = config.min_vel_y;
limits.max_vel_theta = config.max_vel_theta;
limits.min_vel_theta = config.min_vel_theta;
limits.acc_lim_x = config.acc_lim_x;
limits.acc_lim_y = config.acc_lim_y;
limits.acc_lim_theta = config.acc_lim_theta;
limits.acc_lim_trans = config.acc_lim_trans;
limits.xy_goal_tolerance = config.xy_goal_tolerance;
limits.yaw_goal_tolerance = config.yaw_goal_tolerance;
limits.prune_plan = config.prune_plan;
limits.trans_stopped_vel = config.trans_stopped_vel;
limits.theta_stopped_vel = config.theta_stopped_vel;
// Call reconfigureCB function of the planner_util_ object with updated limits and restore_defaults flag
planner_util_.reconfigureCB(limits, config.restore_defaults);
// Call reconfigure function of the dp_ object with updated configuration
dp_->reconfigure(config);
}
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