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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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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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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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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12 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
83 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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13 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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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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23 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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13 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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14 views

queing model M/M/C [closed]

IF X(T)= number of customers in the system at time T in an M/M/C model with C=3 with arrival rate lambda and service rate meu both positive. Then is it true that X(T) is the birth and death processes ...
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35 views

The Hessian of the maximum of a function

I have an optimization problem where the inequality constraint is in the form of: $\min\limits_x f(x)$ subject to $\max\limits_{\zeta} g(x,\zeta) \leq 0$ where $x$ is the optimization variable. I ...
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1answer
18 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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8 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
30 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
15 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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45 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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26 views

How are the penalties derived in the objFcn? (Cell Tower Optimization Problem)

This is the objective function for Demo script for Cell Tower Optimization Problem (This demo illustrates how you would set up and solve an optimization problem (constrained non-linear ...
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35 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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24 views

Making projected search directions conjugate

I'm trying to implement a minimization process for the optimization problem: ...
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1answer
29 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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19 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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50 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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41 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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21 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
44 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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55 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 ...
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19 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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22 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 ...
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3answers
38 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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31 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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[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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70 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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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 ...
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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 ...
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59 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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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) ...
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1answer
21 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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22 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
59 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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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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15 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*} ...
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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. ...
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37 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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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) ...
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2answers
33 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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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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28 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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9 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 ...
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21 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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43 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 ...