There is problem which I am stuck in that for nearly a month.I have encountered a problem which is as follows (equation (1)).\begin{align}&\min_x &f(x) &\quad\implies \text{represents quadratic objective function} \\&\text{subject to } &h_i(x) = 0 &\quad\implies \text{set of $i$ linear equality constraints} \\& &g_i(x) = 0 &\quad\implies \text{set of $i$ quadratic equality constraints}\end{align}I have expected to solve that by iterative algorithm such as SQP (or interior point).I go for SQP but my major problem is divergence matrix of $\{L\}$, i.e. $\nabla^2L$ is singular or non-invertibility (written as formulations below).$$L(x,\lambda,\mu) = f(x) + \lambda^T h(x) + \mu^T g(x)$$$L$ is Lagrangian matrix which includes constraints or maybe an additional penalty function added to constraints for more stability. It is notable that $\lambda$, $\mu$ are Lagrange multiplier, and $k$ denote iteration numbers. The following iterative solution associated with SQP is denoted as below:$$\left[\array{x_{k+1}\\ \lambda_{k+1}\\ \mu_{k+1}}\right] = \left[\array{x_{k}\\ \lambda_{k}\\ \mu_{k}}\right] - (\nabla^2L_k)^{-1} \nabla L_k.$$Recall $\nabla L = \left[\array{\frac{dL}{dx} \\ \frac{dL}{d\lambda} \\ \frac{dL}{d\mu}}\right] = \left[\array{\nabla f + \lambda \nabla h + \mu \nabla g^* \\ h \\ g^*}\right] $Then $\nabla^2L = \left[\array{\nabla^2_{xx}L & \nabla h & \nabla g \\\nabla h & 0 & 0 \\\nabla g & 0 & 0 \\}\right]$How can I employ the SQP method for equation (1) above with this big singularity problem?Is there any modification to SQP for this case to be utilized for problems such as this?Does interior methods suite equation (1) solution and singularity problem.
Iterative optimization problem
optimization;convex optimization;constrained optimization