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 ...

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1answer
44 views

Eliminate cases before calculting all KKT conditions

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-3)^2 + (y-2)^2 \\ s.t. & x^2 +y^2 \leq 5 \\ & x+y\leq 3 \\ & x \geq 0\\ & y\geq 0 ...
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37 views

Non-linear least squares and Bundle Adjustment

In METHODS FOR NON-LINEAR LEAST SQUARES PROBLEMS, 2nd Edition, April 2004 by K. Madsen, H.B. Nielsen, O. Tingleff on page 17 it states: Given a $f: R^n \mapsto R^m$ with $m \geq n$ We want ...
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1answer
47 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
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0answers
16 views

For which values of $c_1, c_2$ and $c_3$ is (1, 2, -2) a local minimum

Consider the problem $$\left\{\begin{matrix} \min & x^2 -2xy + 2xz +y^2 + 4yz + z^2 + c_1x + c_2y + c_3z \\ s.t & g(x,y,z)=-x^2 -4xy - 4xz -2y^2 -4yz - 2z^2 + x -y+z+4 =0 \\ \; & ...
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1answer
20 views

Constrained Optimization: $\min x_1$

Consider the problem $$\left\{\begin{matrix}\min & x_1 \\ s.t & x_2 \geq 0 \\ \; & x_2 \leq x_1^3 \end{matrix}\right.$$ It is asked to find the minimum and show why this does not satisfy ...
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4answers
90 views

Minimization on compact region

I need to solve the minimization problem $$\begin{matrix} \min & x^2 + 2y^2 + 3z^2 \\ subject\;to & x^2 + y^2 + z^2 =1\\ \; & x+y+z=0 \end{matrix}$$ I was trying to verify the first ...
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0answers
23 views

Convex signal reconstruction for convex generator function?

Let $f : \mathbb{R} \mapsto \mathbb{R}$ be a convex (not affine) function and suppose that $y = f(x)$. We want to reconstruct the input $x$ for a given $y$, and the standard approach would be to ...
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0answers
51 views

What curved ramp transports a ball from (1,1) to (0,0) most quickly, under the acceleration of gravity, with no friction or air resistance?

An infinitisemally small ball is placed at the top of a ramp which has a height of 1m and ends 1m away horizontally. What is the optimal curve of the ramp to minimize time taken for the ball to reach ...
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0answers
50 views

Convex envelopes of bivariate functions

In order to convexify my nonlinear non-convex program I need convex envelopes for the function $(x/y)^2$, both x,y are positive. I am only aware of the convex envelopes of the type $xy$ from here ...
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15 views

Is there any non-linear optimization technique whose running time depends on the diameter of the underlying polytope(induced by constraints)

It is well known that the running time of the simplex algorithm depends on the diameter of the polytope induced by the constraints. Is there any non-linear optimization technique that also has this ...
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1answer
47 views

Optimization: maximizing nonconvex sum of product of constraints

I'm wondering if there is any way to convexify, approximate, and/or simplify the following problem. $\max. \sum_{k \in K} \prod_{i \in I} (a_{ik} x_{ik} + b_{ik})$ s.t. $x_{ik} \in [0,1]$ where ...
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0answers
16 views

How to generate feasible $H$-conjugate descent search directions in convex subset

If we want to minimize a quadratic function $f(x)=c^Tx+\frac12x^THx$ (where $H$ is a symmetric positive-semidefinite matrix) in a convex subset $C\subset\mathbb{R}^n$, then is it possible to generate ...
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1answer
248 views

How to linearize this constraint a summation of a product of a integer with a binary

I have to linearize the following constraint, $$ \sum_{i \in V_C} \sum_{j \in V} \sum_{k \in K} y_{ik} \cdot x_{ijk\ell} \leq I_\ell \qquad \forall \ell \in V_D $$ where $y$ is a integer variable ...
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1answer
22 views

Stability of Model Predictive Controller

I appreciate if you suggest a reference that discusses the stability of Model Predictive Controller for continuous-time nonlinear systems.
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0answers
62 views

Polynomial roots finding algorithm

My initial problem is a parameter estimation problem that is solved by minimining a least-square criterion with the Gauss-Newton algorithm. However finding a good initial iterate is very tedious. ...
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0answers
70 views

Solve constrained system of linear equations from samples of a reference function

I have a system of $2n$ linear equations in $2n$ unknowns represented by the standard matrix equation: $$Ax = b$$ Where the solution vector $x = (p_1, ..., p_n, q_1, ..., q_n)$ represents real ...
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0answers
33 views

Making projected search directions conjugate

I'm trying to implement a minimization process for the optimization problem: ...
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1answer
101 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 ...
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0answers
61 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 ...
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0answers
58 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: ...
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1answer
120 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 ...
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0answers
54 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 ...
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1answer
79 views

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

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$
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0answers
155 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
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0answers
47 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 ...
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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 ...
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1answer
122 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 ...
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0answers
72 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 ...
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2answers
99 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 ...
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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 ...
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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 ...
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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. ...
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1answer
198 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 ...
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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), ...
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0answers
198 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)= ...
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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 ...
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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: $$\lambda^*=\underset{\lambda\in \Lambda}{\text{arg max}}\;\;\mathcal{D}(\textbf{x}+\lambda\textbf{d}),$$ ...
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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 ...
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0answers
32 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 ...
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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, ...
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1answer
222 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 ...
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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, ...
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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 ...
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0answers
79 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 $A(x,y), a(x,y), q(x,y)$ ...
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0answers
25 views

Plotting Non Linear Programming functions

I define these functions in Matlab: ...
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1answer
83 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 ...
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1answer
271 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
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3answers
301 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
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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
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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 ...