# Bearing Capacity of Karst Cave Roof under File Foundation Load Using Limit Analysis
## 4 Solution  
The feasible region of the original linear programming problem is a convex polyhedron, therefore, the solution, satisfying the Kuhn-Tucker condition, must be the optimal solution of the original linear programming problem. Note the constraint equation of Equation (24)  
\[\left\{ \begin{array}{ll}\mathbf{B}u-\sum\limits_{\epsilon=1}^{E}\lambda^{ \epsilon}\
abla f^{\epsilon}(\sigma)=0;e=(1,\cdots,\text{N}_{1})\\ \lambda^{\epsilon}f^{\epsilon}(\sigma)=0;\quad\quad e=(1,\cdots,\text{N}_{1}) \\ f^{\epsilon}(\sigma)\leq 0;\quad\quad e=(1,\cdots,\text{N}_{1})\\ \lambda^{\epsilon}\geq 0;\quad\quad e=(1,\cdots,\text{N}_{1})\\ \end{array} \right. \tag{25}\]  
is the Kuhn-Tucker optimality condition for the nonlinear programming problem (26)  
\[ \begin{array}{ll}\text{Maximize}&Q=\sigma^{\text{T}}\mathbf{B}u+\mathbf{c} _{u}^{\text{T}}u+\mathbf{c}_{v}^{\text{T}}v(on\text{ \it stress field})\\ \text{Subject}&\text{to}&f^{\epsilon}(\sigma)\leq 0;e=(1,\cdots,\text{N}_{1}) \\ \end{array}  \tag{26}\]  
Simultaneous Equations (24) and (25) obtain a nonlinear programming problem of the double objective function, which can simultaneously solve the stress field and velocity field of the karst foundation in the critical state. \[ \begin{array}{ll}\text{Maximize}&Q=\sigma^{\text{T}}\mathbf{B}u+\mathbf{c} _{u}^{\text{T}}u+\mathbf{c}_{v}^{\text{T}}v(on\text{\it stress field})\\ \text{Minimize}&Q=\sigma^{\text{T}}\mathbf{B}u+\mathbf{c}_{u}^{\text{T}}u+ \mathbf{c}_{v}^{\text{T}}v(on\text{\it velocity field})\\ \text{Subject}&\text{to}&f^{\epsilon}(\sigma)\leq 0;e=(1,\cdots,\text{N}_{1}) \\ \end{array}  \tag{27}\]  
Writing the constraint \(v^{d}\geq 0\) as \(f(v^{d})\leq 0\), the Kuhn-Tucker optimality condition of Equation (27) can be expressed as  
\[\left\{ \begin{array}{ll}\mathbf{B}u^{*}-\sum\limits_{\epsilon=1}^{E}\lambda _{\sigma}^{E}\
abla f^{\epsilon}(\sigma^{*})=0;e=(1,\cdots,\text{N}_{1})\\ \mathbf{c}_{v}+\left(\mathbf{A}_{v}^{d}\right)^{\text{T}}\mathbf{\mu}+\sum \limits_{\epsilon=1}^{E}\lambda_{v}^{\epsilon}f^{\epsilon}(\sigma^{*})=0;e=(1,\cdots,\text{N}_{1})\\ \mathbf{B}^{\text{T}}\sigma^{*}+\mathbf{c}_{u}+\mathbf{A}_{u}^{\text{T}}\mathbf{ \mu}=0\\ \mathbf{A}_{u}^{d}u^{d}+\mathbf{A}_{v}^{d}v^{d}=\mathbf{0};d=(1,\cdots,\text{N}_ {2})\\ \mathbf{A}_{v}^{\epsilon}\
abla f^{\epsilon}(\sigma^{*})=0;e=(1,\cdots,\text{N }_{1})\\ \lambda_{\sigma}{}^{\epsilon}f(\sigma^{*})=0;d=(1,\cdots,\text{N}_{2})\\ f^{\epsilon}(\sigma^{*})\leq 0;e=(1,\cdots,\text{N}_{1})\\ f^{\epsilon}(\sigma^{*})\leq 0;e=(1,\cdots,\text{N}_{1})\\ f^{\epsilon}(\sigma^{*})\leq 0,\lambda_{v}^{\epsilon}\geq 0\end{array} \right. \tag{28}\]  
where \(\lambda_{\sigma}^{\epsilon}\), \(\lambda_{\sigma}^{\epsilon}\) are the non-negative plastic multipliers on the velocity and stress fields of each element, respectively, \((\mathbf{u}^{*},\mathbf{v}^{*},\sigma^{*})\) is the optimal solution of the original linear programming problem, and \(\mathbf{\mu}\) is a coefficient vector for the discontinuity variables. Following the discussion above, the initial nonlinear programming problem is converted into a series of Kuhn-Tucker optimization conditions, which are subsequently resolved via the two-stage quasi-Newton algorithm. The fundamental premise of the quasi-Newton and Newton methods is to linearize the nonlinear equation by employing the Taylor expansion at the iteration point. During each iteration process, the iteration direction is aligned with the decrease in function value at the current point. The iteration continues until the minimum value of the objective function, which fulfills the accuracy requirement, is obtained. However, it should be noted that the Newton method necessitates the calculation of the Hessian matrix or inverse matrix during each iteration, which is time-consuming and may also be challenging or infeasible to compute. In contrast, the quasi-Newton method employs a positive definite symmetric matrix to approximate the Hessian matrix or its inverse matrix, facilitating a more manageable resolution of the problem.