5

This is just an NLP (Non-Linear Programming) model. You can rewrite it as: $$\begin{align}\min \>& Z \\ & \sum_i w_i = 1 \\ & \sum_i f_i(w_i\cdot Z)\ge k\cdot Z \\ & w_i \in [0,1] \end{align}$$ Getting rid of a division is always a good idea. If we can assume $Z\gt0$, then a slightly different formulation can look like: $$\begin{align}\...


4

In SCP (a.k.a. SCA), at each outer iteration: Objective function is replaced by a convex approximation, not necessarily quadratic. Nonlinear inequality constraints are replaced by convex approximations, not necessarily linear. Nonlinear equality constraints are replaced by linear approximations. Therefore, at each outer iteration of SCP, a convex ...


4

First, a standard semidefinite program (in primal form) would be $$\underset{X}{\text{min }} \langle W, X \rangle, \\ X\succeq 0,~\mathbf{A}(X) = b$$ where $\mathbf{A}(X) = b$ denotes the primal equalities (the model you have written is either trivially solved by $X=0$ or is unbounded) In reality one would never work with such a limited form, so you can at ...


3

For debugging purposes, it's helpful to identify small subsets of infeasible constraints in an LP formulation- an Irreducible Infeasible Subset (IIS) is a subset of the constraints that are infeasible and such that removing any constraint from the IIS results in a feasible set of constraints. This is doable in polynomial time and fairly fast in practice, ...


2

We can do some transformations to your problem to show that it's easily solvable via a linear program: Given a matrix $M$ with non-negative real entries and a vector $v$ you wish to solve the problem: $$ \begin{align} \min_v \quad & \lVert Mv \rVert_\infty \\ s.t. \quad & v_i\ge0 \\ & \sum_i v_i = 1 \end{align} $$ Now, note that $\lVert Mv \...


2

With your constraints of $a = 1$ and $c = -\frac{b}{d} - 1$, your $f$ looks like this: $$\hat{\mathbf{y}} = f(\mathbf{x},b,d) = \frac{bd}{d\exp{(\mathbf{x})}-b - d} + d$$ You're trying to solve a nonlinear least square problem basically: $$b,d = \arg \min \sum_{i=1}^{N} (f(x_{i})-y_{i})^{2}$$ It's a job for scipy.optimize.curve_fit: import numpy as np from ...


2

The two solvers that you've mentioned both implement branch and bound methods for integer linear programming. The differences are at the implementation level rather than in the basic algorithms being used.


2

That can't happen. You can have more constraints than variables, but the number of active constraints can not exceed the number of variables. (That's not quite true: Some constraints might be redundant, or degenerate, but then you can remove these constraints from the active set with no ill effect.) Why my statement above is correct is probably easiest to ...


1

I am not sure if this is going to work as you want -because there is very little information in your question-. But you can introduce a vector of binary variables $w\in\{0,1\}^n$, then add $x_i = w_i\times b_i$ -where $b_i$ is a real number-, and $\sum_i w_i = 1$ (summation in the classical sense not XOR, e.g. $1+1=2$) as constraints. The second condition ...


1

There has been progress both in terms of the speed of computers (basically driven by Moore's law) and in the algorithms used to solve LP's and especially MILP's. Overall, improvements in algorithms have had at least as large an effect as improvements in hardware. How this will work out for your particular model is a different question. Especially for MILP'...


1

NLopt has about a dozen local derivative-free optimizers, including SLSQP in C. Only COBYLA currently supports arbitrary nonlinear inequality and equality constraints; the rest of them support bound-constrained or unconstrained problems only. (However, any of them can be applied to nonlinearly constrained problems by combining them with the augmented ...


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