5

Use CVX's entr function. $\sum_{i=1}^ 4x_i\ln(x_i)$ can be entered as -sum(entr(x)) entr Scalar entropy. entr(X) returns an array of the same size as X with the unnormalized entropy function applied to each element: { -X.*LOG(X) if X > 0, entr(X) = { 0 if X == 0, { -Inf otherwise. If X ...


3

The gradient and Hessian of $f$ at $x=x_0$ are constants (vector and matrix respectively), so $f_0(x)$ is a quadratic function of $x$. If it helps, write it as $f_0(x) = c_0 + c^Tx + x^T D x$. Hopefully you know the gradient of that. Then figure out what $c_0$, $c$ and $D$ are in terms of $x_0$, $f(x_0)$, $\nabla f(x_0)$ and $Hf(x_0)$, substitute and maybe ...


3

Note that by assumption, there must be some $i \in I(x^*) \cup \{0\}$ with $y_i \ne 0$. Let $\mathcal I = \{i \in I(x^*) \cup \{0\} \mid y_i \ne 0\}$ be the set of such indices. Let $u$ be any unit vector of the same dimension as $x^*$. We wish to show that $x^* + tu$ is either not feasible or has weakly worse objective for any positive scalar $t$. To show ...


2

Binary (Boolean) values are integer values. Therefore, optimization problems with boolean constraints are either integer programming or mixed integer programing (MIP). Generally, there is no easy algorithm that is guaranteed to find the optimal solution of MIP problems quickly. My reason to believe that is: I can implement a boolean satisfiability (b-sat) ...


1

This sounds like parametric programming to me. In short, you can calculate a set of regions, typically called "critical regions", within which the same set of constraints is active. This in return enables you to calculate $x$, $\lambda$ and $\mu$ as an explicit function of your parameter $a$. There are MATLAB tools to do this, most prominently the ...


1

Minimize $x^2$ where $1 \le x \le 2$. \begin{aligned} \min_{x} \quad & f(x)\\ \textrm{s.t.} \quad & h_{1}(x) \le 0\\ &h_{2}(x) \le 0 \\ \end{aligned} where \begin{align} f(x) &= x^2 \\ h_{1}(x) &= 1 - x \\ h_{2}(x) &= x - 2 \end{align} KKT conditions: \begin{align} 0 &= \nabla f(x) + \mu_{1}\nabla h_{1}(x) + \mu_{2} \nabla h_{...


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