A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for ...

learn more… | top users | synonyms

0
votes
0answers
32 views

Making projected search directions conjugate

I'm trying to implement a minimization process for the optimization problem: ...
0
votes
1answer
98 views

KKT conditions for a convex optimization problem with a L1-penalty and box constraints

I am having some trouble deriving / understanding optimality conditions for a convex optimization problem of the form: $$\begin{align} \min_{x\in\mathbb{R}^d}~ & f(x) + C.\|x\|_1 &\\ &x_i ...
0
votes
0answers
58 views

How to obtain the optimal lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found.?

I am using a blackbox solver to solve the following nonconvex QCQP to global optimality. $$ \min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u $$ where $Q_0$ is ...
0
votes
0answers
56 views

Linearisation of nonlinear system

I'm asked to write the linearisation of this nonlinear system around the equilibria: $$\begin{cases} x_{t+1}=-x_t+2x^2_t \\ y_{t+1} =-2x_t^2-y_t\end{cases}$$ The two equilibria are therefore: ...
0
votes
1answer
114 views

SDP relaxation of a non-convex quadratically constrained quadratic program.

I am very new to SDP and SDP solvers. I have a semi definite program of the following form $$\min_{x,X}\ Q\bullet X+c^Tx$$ $$\text{s.t. } Q^k \bullet X + (c^k)^T x =b^k , \ k=1,2, \dots,m \\ \quad ...
0
votes
0answers
52 views

When is a quadratically constrained quadratic program (indefinite objective matrix) unbounded?

I have a nonconvex QCQP of the form $$x^TQ_0x + c^T x$$ such that $x^TQ_1x+c_1^Tx=b_1$, $Ax=b$, and $l\leq x\leq m$ where $Q_0$ is indefinite diagonal matrix and $Q_1$ is positive semidefinite ...
1
vote
1answer
78 views

Minimize $w=9y_1+4y_2$ subject to linear inequalities

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$
3
votes
0answers
150 views

Jackson's theorem to optimize mean queue length of a traffic model

I am working on traffic signals for a city transport system. I modeled the city transport using a queuing network as shown in the following image Arrival rate of "A" cars from outside is S1 and ...
2
votes
0answers
44 views

Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta ...
0
votes
0answers
41 views

solving non-linear system of equation as optimization problem

I am trying to solve a system of non-linear algebraic systems through optimization. $f_i(x)=0$ for $i=1..n$ and $x \in R^n$. I saw few optimization versions: Minimize $\sum f_i^2(x)$, Minimize ...
1
vote
3answers
48 views

Constrainted optimization involving logarithms

The problem is to minimize $ f(x_1, x_2 ,x_3, x_4):= - \Big[ \log ({\frac{1}{4} + x_1}) + \log ({\frac{1}{2} + x_2})+ \log ({\frac{1}{5} + x_3})+ \log ({\frac{3}{4} + x_4}) \big]$ such that ...
0
votes
1answer
113 views

Dynamic programming approach for multidimensional problem

I use a dynamic programming approach to optimize the behaviour of individuals playing a game.I have one strategy matrix that describes the behaviour of individuals in situation 1, which depends on ...
3
votes
0answers
70 views

[Levenberg-Marquardt]What is the link between positive-definiteness and well-conditioning?

Working on optimization problems through neural networks, I use the Levenberg-Marquardt algorithm. I have read this assertion that I do not understand : A positive definite diagonal matrix is ...
5
votes
2answers
97 views

General solution to a system of non linear equations with a specific pattern

I am seeking a general solution to a system of non linear equations with a specific pattern: Order 1: $$ x_0 = a^2 + b^2 $$ $$ x_1 = 2ab $$ Order 2: $$ x_0 = a^2 + b^2 + c^2 $$ $$ x_1 = 2ab + 2bc ...
0
votes
0answers
33 views

how to solve this exponential optimization problem?

Let $v_{ik}=\log ({\lambda _{ik}}) - {\lambda _{ik}}{T_k}$, $s_{ik}=\log (\frac{1}{C-\sum_{k=1}^{K}{N_k}}) - {\lambda _{ik}}{T_k}$. Consider the following optimization problem: $$ \text{minimize ...
0
votes
0answers
75 views

Solving the quadratic optimization problem with quadratic inequality constraint

I have a quadratic optimization problem which which both objective function and constraint are convex. As the problem is very big, I used decomposition technique and divide the problem to smaller ones ...
0
votes
0answers
87 views

Solving constrained linear programming problem

For the variable $t$, problem is to find best multipliers $k$ which minimizes the objective function. Time: $t_1$, $t_2$, $t_3$,... given in input Multiplier $k_1$, $k_2$, $k_3$,... (These are ...
0
votes
0answers
25 views

Reducing uncertainty of a mathematical model with data (Process control)

I know this is a very broad question, but need suggestions, link to good reference papers etc. So here is the question: I have an uncertain model whose parameters are static (not changing with time) ...
0
votes
1answer
31 views

Explanation of $\textrm{argmax}_j{z_jt}$ and how to implement it

I'm struggling with one equation within a subtour elimination constraint. $$\sum_{i \in S} \sum_{j \in S, j<i} y^t_{ij} \le \sum_{i \in S} z_{it} -z_{kt} \quad S \subseteq M \quad t \in T \quad ...
0
votes
0answers
31 views

joint optimization problem with somewhat symmetric function

I have just brief question that the method that I use to solve optimization problem is legit. I have function $\max_{x,y}F(x,y)$, and first order condition gives me following equation. ...
0
votes
1answer
190 views

MATLAB: minimize function using x value from previous iteration

I'm trying to develop an algorithm for a proximal point method defined as: $$ \underset{x \in \rm I\!R^n}{\arg\min} f(x) + \lambda g(x) $$ where f(x) is a convex and coercive function and also ...
0
votes
0answers
24 views

Find $\alpha$ from equation $F(\alpha)=\int \left (\frac {I(x)}{\alpha^TG(x)}-1\right)^2 \, dx+\lambda\|\alpha\|^2$

I have a function such as $$F(\alpha)=\int \left (\frac {I(x)}{\alpha^TG(x)}-1\right)^2 \, dx + \lambda \|\alpha\|^2$$ where $I,\lambda,G$ are given. In which $G(x)$ is a vector; $G=[G_1(x), G_2(x), ...
0
votes
0answers
22 views

Help required in solving the lagrangian dual?

I'm trying to write the Lagrangian dual to the following problem \begin{align*} (P) \quad \min\;&\text{Trace}(CG)\\ \text{s.t.}\;&G \succcurlyeq 0\\ & G_{i,i}=I_d (i=1,..,M+1)\end{align*} ...
0
votes
0answers
10 views

If $d^k$ is a descent direction for $f(x^k)$, then $\nabla f(x^{k+1})^t d^k=0$, where $x^k+1=x^k+\alpha^k d^k$

I have solved this problem but got something slightly different. My professor is notorious for his typos in half of the examples, so I was wondering if this is just another typo or if I got it wrong. ...
0
votes
0answers
181 views

Convert a nonconvex function to convex function

I have a image $I: \Omega \to \Bbb R$. It is separated into 2 non-overlapping region: $D$ and $\Omega \setminus D$ Each point $x$ in the image $I$, the $\phi$ function is defined as: $$\phi(x)= ...
0
votes
0answers
37 views

Justifying the “Dual feasibility”, one of the Karush-Kuhn-Tucker conditions

I am having difficulty of interpreting the KKT conditions in a general setting where we have $M$ equality and $N$ inequality constraints defined as: Minimize $f(x)$ subject to $g_i(x) \leq 0 , h_j(x) ...
0
votes
2answers
41 views

Quite rare kind of proof of convexity for a quadratic function!

Excuse me all of you in advance. I got this problem as an assignment but I am not really good doing proofs! If $f(x)=\frac12x^TQx+b^Tx+a$ is quadratic in $n$ variables, where $Q$ is symmetric. Show ...
1
vote
1answer
45 views

Why does $\frac{\textbf{g}^T\textbf{d}}{\textbf{d}^T\textbf{H}\textbf{d}}$ give the maximum of function $\mathcal{D}(\textbf{x}+\lambda\textbf{d})$

Let say I have to find the value $\lambda^*$, that maximizes the following quantity: $$\lambda^*=\underset{\lambda\in \Lambda}{\text{arg max}}\;\;\mathcal{D}(\textbf{x}+\lambda\textbf{d}),$$ ...
0
votes
1answer
30 views

What is the approach to solve simple constrained optimization when first order condition $\nabla f = 0$ yields solution outside of domain

I wish to solve the problem min $ f(x_1, x_2) = x_1^2 - x_1 + x_2 + x_1x_2$ subj $x_1\ge 0, x_2 \ge 0$ We find $\nabla f = [2x_1 - 1 + x_2, 1+x_1] = 0$ yields $x_1 = -1, x_2 = 3$ which is outside ...
0
votes
0answers
12 views

Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$

I have been stuck at this problem for a while :( Given $\mathbf{A}\in\mathbb{S}^{p\times p}, \mathbf{A}\ge 0,\mathbf{A} \text{ symmetric}, \mathbf{b}\in\mathbb{R}^n,\mathbf{c}_i\in\mathbb{R}^p\forall ...
1
vote
0answers
31 views

Why steepest descent gives a wrong direction search?

I have to minimize the function $ƒ(x_1,x_2)=(x_1-1)^2+x_2^3-x_1x_2$. The initial point is $[1,1]^T$. The gradient of this function is $∇ƒ(x_1,x_2)=[2(x_1-1)-x_2,3x_2^2-x1]$. This gradient evaluated ...
0
votes
0answers
101 views

Karush-Kuhn-Tucker conditions for non-linear optimalization

I have the following problem: solve the local conditions (KKT) and find ALL optimal solutions: $$\min f(x,y)$$ subject to $$g(x,y)\le 0$$ $$x\geq0, y\in\mathbb{R}$$ I have some questions to this ...
1
vote
1answer
48 views

(half) hyperboloid least squares problem

I have five equations as follows, , where i = 1, 2, 3, 4, 5 and only (x, y, z) are unknown. The five equations above are half-side hyperboloids. It could be seen as . I want to find the solution (x, ...
1
vote
1answer
206 views

How to project gradient vector to subspace defined by linear constraints

I have the following set of linear constraints: $$\begin {align}\textbf{y}^T\textbf {x} &= 0 \\ \textbf {0} &\leq\textbf {x} \leq C\cdot\textbf {1},\end {align}$$ where $\textbf {y} \in ...
0
votes
0answers
32 views

what kind of optimization is this?

I have an optimization problem that looks like this: \begin{array}{cc} min & x'\varSigma^{2}x+k^{2}e'e-2ke'\varSigma x\\ s.t. & x'\varSigma x=ke'x\\ & ...
0
votes
1answer
64 views

How to find Parameters in nonlinear Regression Model?

I have a nonlinear Regression Model with eleven observations of $x,y$. How do I find the parameters $a,b,c,d$ of the model: $ y=f(x)=a + b \sin cx e^{dx}$ by using the function: $$\Phi(a, b, c, ...
0
votes
0answers
18 views

How to find parameters from logistic equation

I have an function and assume that that is convex function. I want to use gradient decent to find parameters in that equation. Could you suggest to me the way to do it. Thanks. This is my function ...
1
vote
0answers
74 views

Optimization with Integral Inequality constraint and nonnegativity conditions

Trying to solve this: $$\min TC(A,a,q)= \int_M f(A,a,q)\,dx\, dy$$ $$s.t.$$ $$a\le\int_M g_i(A,q)\,dx\,dy$$ $$q\le \text{constant}$$ $$A,a,q\ge0$$ $(x,y)$ is omitted in $A(x,y), a(x,y), q(x,y)$ ...
0
votes
0answers
25 views

Plotting Non Linear Programming functions

I define these functions in Matlab: ...
0
votes
1answer
80 views

How to determine the optimal step size in a quadratic function optimization

I have the following optimization problem: $$\underset{\alpha\in\mathbb{R}}{\text{min}}:\;\;f(\textbf{x}+\alpha\textbf{d})$$ $$\text{subject to}:\;\;0\leq\alpha\leq \alpha_{max},$$ where ...
1
vote
1answer
255 views

How do I convert a constraint with a product of two integer variables to a linear constraint?

I have a constraint of the form: $$\theta \leq a_1x_1 + a_2x_2 + a_3x_1x_2$$ where, $x_1$ and $x_2$ are integer variables with ranges $x_1 \in \{0, m\}$ and $x_2 \in \{0, n\}$. I would want to ...
1
vote
3answers
284 views

ADMM formalization

I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and ...
1
vote
0answers
43 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
0
votes
1answer
39 views

How to find optimized value of two variables

I have two variables: $\kappa_y$ and $\kappa_x$ And three functions: $M_y$=$M_y$($\kappa_y$, $\kappa_x$) $M_x$=$M_x$($\kappa_y$, $\kappa_x$) $F_z$=$F_z$($\kappa_y$, $\kappa_x$) All these three ...
2
votes
2answers
177 views

Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
1
vote
0answers
45 views

Why Was Backprop Invented?

I'm currently researching artificial neural networks and I keep wondering why do we use "backpropagation" to train a neural network. An ANN is basically just a very large and complex function ...
1
vote
1answer
31 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
1
vote
1answer
47 views

Gauss-Newton Non-Linear Squares Optimisation

I doubt this is solvable at all, but I thought I will give a try. Essentially I am trying to extend Gauss-Newton algorithm to 2nd Taylor term. ...
0
votes
1answer
26 views

How do I know that method of steepest descent works?

Here is the definition of the method of steepest descent given in the book "The mathematics of nonlinear programming" by Peressini. Suppose $f(x)$ is a function with continuous partial derivatives on ...
0
votes
0answers
35 views

Global stochastic maximization of a multi-parameter function

I have a function $F:\mathbb{R}^n\to[0,1]$ such that $$ F(\lambda) = \mathbb{E}_x[f(\lambda;x)] = \int f(\lambda;x)\mu(x)dx,$$ and I want to find $\tilde\lambda$ that maximizes F, i.e. ...