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# Questions tagged [nonconvex]

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• 141
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### Is there an optimization scheme/algorithm that converges, for this non-convex scenario but with some special properties

I have a smooth function $f(x) = \frac{g(x)}{h(x)}$ that is the ratio of two smooth convex functions $g(x)$ and $h(x)$. It is known that $f(x)$ has a global minimum, achieved at the unique point $x_0$....
• 141
2k views

### A robust algorithm to sort a non-convex polygon vertices

Let v_{0},...,v_{N-1} be N points in a Cartesian xy plane defining the vertices of closed polygon (i.e. v_{N} = v_{0}). Let P_{0}...
• 139
1 vote
52 views

• 113
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### reduced system: primal-dual interior point method for nonconvex constrained problem

When solving a reduced KKT system of a nonlinear (and nonconvex) constrained program after eliminating slack and dual variables, how do we actually take the next step in a primal-dual method? For ...
• 325
89 views

### Why can't we discretize continuous domains in distributed non-convex constraint optimization problems?

Consider a non-convex distributed optimization problem. We have $X$ = a set of $n$ decision variables: $x_i$ where $i=1..n$ and $x_i \in R$, the set of Reals. We have $F$ = a set of $m$ constraint ...
1 vote
110 views

### Minimizing the products of variables

My problem Maximize $$\min_{i} \{\ c_i \cdot \prod_{j \in A(i)} {x_{j}} \prod_{j \in B(i)} {y_{j}} \}$$ Subject to \begin{align} &\sum_{j \in C(k)} x_{j} = 1,\ \forall k \\ &l \leq x_{j}...
• 111
1 vote
293 views

### Constrained optimization: Stationary point vs. Nash point

1s question: definition of stationary point for constrained optimization As far as I know, a stationary point of a constrained optimization problem is a stationary point of the Lagrangian (that has ...
• 515
180 views

### Optimization of non-smooth, non-convex, locally Lipschitz functions of type exp(-abs(x))

What would be the numerical method of choice to find minima in a non-smooth, non-convex, locally Lipschitz function $f: \mathbb{R}^n\rightarrow \mathbb{R}$. The function $f$ is mostly smooth but ...
• 193
90 views

### Global optimization with known distributions of some variables

I'm solving simple single-objective multidimensional global optimization problem using various stochastic algorithms like Monte-Carlo, GA and other evolutionary approaches. The task is formulated as ...
• 21
1 vote
174 views

### preconditioning LBFGS?

I want to minimize an energy of the form $$V_1(\mathbf{x}) + V_2(\mathbf{x})$$ where $V_1$ is much stiffer than $V_2$. When I try to use LBFGS, convergence is extremely slow, as the solution ...
• 263
677 views

### Global convergence in trust region algorithm

I was reading about TR methods and there are some terms, which are confusing for me. It says, method is globaly convergent. What does it really mean? Converges to global minima, or converges for ...
• 109
1 vote
281 views

### Numerical solution of non-linear advection equation other than inviscid burgers

I am solving a non-linear advection equation of the form $u_t + f(u)_x = 0$ where $f(u)$ is a complicated function of $u$. I am solving this equation using a first order fully implicit scheme (...
• 23
10k views

### Why should non-convexity be a problem in optimization?

I was very surprised when I started to read something about non-convex optimization in general and I saw statements like this: Many practical problems of importance are non-convex, and most non-...
1 vote
153 views

### Minimization of the sum of convex function and non-linear non convex function

I'm trying to minimize the unconstrained scalar sum of a quadratic convex function (to which a convex optimizer is readily applied) and a non-linear and non-convex function which is differentiable. ...
• 141
4k views

In these notes (section 2.3), it is stated that: A point $x^*$ is a minimizer of a function $f$ (not necessarily convex) if and only if $f$ is subdifferentiable at $x^*$ and $0 \in\partial f(x^*).$ ...
• 515
275 views

Suppose $A\in\mathbb{R}_+^{n\times n}$ is symmetric. I would like to factorize $A\approx UU^\top$ by solving $$\begin{array}{rl} \min_U & \sum_{ij} \left(A_{ij}\ln\frac{A_{ij}}{[UU^\top]_{ij}}+[... • 1,341 2 votes 0 answers 103 views ### non convex, non linear optimization involving matrix differential equation solution I'm trying to develop an inferential procedure for a multivariate dependent Markov process. Basically, the procedure could be considered as a non linear regression, with a known dependence structure ... 2 votes 1 answer 331 views ### Minimal surface finite differences problem - Matlab assemble I face to the following problem:$$(1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0.$$Problem needs to be discretized and assembled. Does anybody know how to proceed in Matlab? 2 votes 0 answers 362 views ### Finite difference scheme for solving nonlinear least-squares problem I am dealing with following problem:$$ \min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma (...
233 views

I have a non-convex optimization problem in the form: \begin{align} \min_{b,\xi,\eta} \sum_{i=1}^{n} b_i \xi_i + \gamma \Vert \eta \Vert \cr \text{s.t.} b\geq 0, b^\mathsf{T} 1 = 1,b_i \leq \frac{1}{n-...
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