Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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Determining initial values for optimization problem

I am trying to solve an optimization problem with a quadratic objective function and non-linear constraints, using SQP (Sequential Quadratic Programming). I am attempting at doing the implementation ...
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23 views

Least squares with a quadratic inequality constraint

Is there a closed form solution for the following least squares problem: $$ \min_\mathbf{x}\|\mathbf{a+Bx}\|^2 ~~\text{s.t}~~\|\mathbf{x}\|^2 \leq \alpha^2$$ where $\mathbf{a} \in \mathbb{C^{M\times ...
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Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
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How to interpret a discontinuity in 2D Pareto Frontier?

I've solved a bi-objective optimization problem by means of NOMAD solver from OPTI Toolbox and as a result I've obtained a Pareto frontier: How to interpret the visible "gap" in the Pareto frontier? ...
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Optimisation: Maximum of a rectangle with semi circles at each end

A field is being built in the form of a rectangle with semi circles at each end. A $400$m racetract to is be built around the playing field. a) What Radius of the semicircular end would give the ...
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How to efficiently compute the pareto front in a >2 dimensional multi-objective case?

I'm currently working on an optimization problem with 4 different objective functions and need an algorithm to compute the pareto frontier from several "solutions" to that problem. I already found ...
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25 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
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1answer
30 views

Constraineed Maximization Problem

I am creating a game, and have run into quite a tricky problem which I have been wrestling for days. I have been able to turn into somewhat mathematical terms (bare with me, I'm a programmer not ...
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1answer
65 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
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1answer
34 views

Global/local optima for this function

I have the following function $f(x_1,x_2) = \frac{x_1}{x_2+p} + \frac{x_2}{x_1+p}$ where $x_1$ and $x_2$ $\in$ $[0,1]$ and $p > 0$ is a constant I want to find global/local maxima for this. ...
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Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
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Method to calculate the best way to repay two different loans given a set amount of money per month?

Given two (or more) loans of different balances and interest rates and a single amount of funds available per payment period, is there a way to calculate the best way to split the available funds to ...
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1answer
21 views

normal equations of $ y(t) = \gamma e^{\lambda t} $ for minimizing the error

Let $ y(t) = \gamma e^{\lambda t} $ and we have the points $(0,2)\ (1,0.7)\ (3, 0.3)$. The task is to get the parameter so that error is minimal. So we need to get the matrix for the normal ...
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2answers
82 views
+100

How exactly do I prove that I find the maximum of the function

I am currently trying to maximize an objective function $f(a,b,c,d,e)$ over the variable $b$ only. By taking the derviative of f over b, setting it to zero, I can solve b in terms of the other 4 ...
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32 views

Minimizing sum of functions implies minimizing their squares, maximizing the sum of the inverses?

I have $n$ functions (Say $f_1\space to \space f_n$) of $k$ variables (Say $x_1\space to\space x_k$) each. The functions are all positive, as well as the variables $xi's$. I do not have explicit ...
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3answers
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Optimization with contraint

Given the value K with constraint x+y = K, what can be the maximum value of x*y be? How did they derive this answer? It is equivalent to finding the maximum value of x*(K-x), which will happen when x ...
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Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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39 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
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1answer
30 views

sum of logarithms of linear-fractional functions Optimization Problem

I am new to optimization theory and I am facing this optimization problem. \begin{equation} maximize \qquad f(x) = \sum_{i} ...
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29 views

multivarable optimization problem, what is the procedure?

Sorry for this obvious question. I am trying to maximize an objective function that consist of 5 variables (a,b,c,d,e) over a and b. That is , $max _{a,b}f(a,b,c,d,e).$ So I procedure I took is ...
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28 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
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1answer
49 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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2answers
86 views

Unsolvable(?) Assignment Problem

I've recently been trying to implement the Hungarian Method in C++, and I've been using 5x5 matrices to test my program. Last night I came across a matrix which neither I nor my program can solve. Is ...
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23 views

Volume of a polytope cut off by a hyperplane

Given a maximization problem with constraints, and adding a few more constraints using the Gomory cuts and solving the relaxed maximization problem, we can arrive at integer solutions. I am looking to ...
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Dimensional Consistency in Grids used in Optimization

I am working on an optimization problem in the research I am doing and my partner and I have found that in order to quickly converge on a solution using a specific PSO (the firefly algorithm - it's ...
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Find unknown such that four dependent quantities have the same value.

I have $12$ unknown $a_i, b_i, c_i, i=1,\ldots,4$, that should satisfy equations $$ \sum_{i=1}^4n_ia_i=a,\quad\sum_{i=1}^4n_ib_i=b,\quad\sum_{i=1}^4n_ic_i=c, $$ where $n_i,\,i=1,\ldots,4$ and $a,b,c$ ...
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Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
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4answers
42 views

Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
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compute a certain maximum in MATLAB

let $ C \in \mathbb{N} $ and $ c_1>c_2>\ldots>c_k \in \mathbb{N} $ with $ C>c_1 $ and $ c=(c_1,c_2,\ldots,c_k)^\top \in \mathbb{N}^k $, where $ \mathbb{N} $ are the natural numbers without ...
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Find scaling factor that minimizes f(x) - round(f(x))?

Let's say I have a function $f(x)$, which has a fractional component $\{ f(x) \} = f(x) - \lfloor f(x) \rfloor$. I would like to add a scaling factor $h(x)$, where $h(x)$ is a polynomial, such that ...
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Hyper-plane that separating hyper-cube.

Suppose $\Omega \in \mathbb{R^4}$ is closed unit ball in $ ||.||_{inf}$ i.e. Hyper-cube. 1) Am I right that there are L=16 extreme points of $\Omega$, all are vertices of the hyper-cube. 2)Is it ...
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A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
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How to introduce flat cost of flow over a node using mixed integer programming.

In the set up for the program we have a graph where we are trying to minimize the cost of sending flow over the arcs. I have formulated the following linear program. \begin{array}{ll} \text{minimize} ...
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Help with Lagrangian Constrained Optimisation

Question: Maximise f (x, y) = x2y, where (x, y) ∈ R2 given the constraint that all (x, y) are points on a circle with radius √3 around origin (0, 0). Solution: f (±√2, 1) = 2 is the maximal value ...
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Unconstrained optimization problem (lasso with modification)

I am looking to solve the following unconstrained optimization problem: $$\arg \min_U \|b-A(UY^*)\|_F^2+\lambda\|U\|_1$$ where $\|.\|_F$ is frobenius norm. I know that the solution without the ...
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1answer
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Optimum set partitioning with constraint

Be $A \subset D \wedge m \in D \wedge \forall x \in A:x < m$, with $D$ finite and included in the positive integers, I need to partition $A$ into $B_n$, while minimizing $n$, so that ...
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2answers
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L1 regularized SVM in Matlab

Minimizing the following SVM formulation \begin{align} \arg\min_{\mathbf{w}}\frac{1}{2}\|\mathbf{w}\|^2_2 \\ \text{subject to } \quad y_i(\mathbf{w}\cdot\mathbf{x_i}) \ge 1 \end{align} can be done ...
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1answer
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Horn–Schunck method. Explanation of iterative solution

I am reading this paper (explanation of Horn-Shunck method for finding optical flow) and trying to understand it. My stumbling block is obtainig solution of system of linear equations I(x, y, t) ...
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1answer
31 views

Reference for gradient descent with unit norm constraint

I faced a non-convex optimization problem with unit norm constraint. I can solve the problem using the gradient descent method and the projection of the gradient onto the tangent plane as in @joriki ...
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1answer
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Max/Min Notation Question

In a paper I'm currently reading it gives alpha to be the following value. $\alpha = \max_t \min_{t_j \in T_N} ||t-t_j||_2$ I am wondering what exactly this means? I have the following code: ...
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multi-objective optimization

I am currently encounterring a optimization problem. The goal is optimize an objective function A and B at the same time. But the problem is that optmizing A will almost always tradoff with B, such ...
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Minimal volume of a tetrahedral

I'm unsure how to solve the following problem: Let $\textbf{p}=(a,b,c) \in \mathbb{R}^{3}$ with $a,b,c > 0$. For $\alpha , \beta > 0$ the equation $$\alpha (x-a)+ \beta(y-b) + (z-c) =0$$ ...
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1answer
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Determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3 metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and front (BC) are ...
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1answer
18 views

Goldstein test in nonlinear programming

I'm reading about nonlinear programming and the Goldstein test. Here is the definition from my book: A line search accuracy test that is frequently used is the Goldstein test. A value of ...
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determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3  metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and  front (BC) are ...
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1answer
23 views

Above what order of magnitude a pure cutting-plane algorithm must be forgotten in favour of branch-and-cut?

Crawling the web on the subject of the cutting-plane algorithm, I have seen everywhere that a pure cutting-plane method cannot be used for numerical instability reasons after some iterations. But do ...
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How to transform a maximizing objective function which contains a max operator to a standard LP form

My Optimization objective function looks like this: $\max\quad(c_1 x_1 + c_2 \max\{x_2, x_3, x_4\})$ all variables, $x_i$ are binary variables. There are also some linear constraints such as $a_ix_1 ...
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1answer
25 views

What is $s$ in s-energy (eg. Riesz s-energy)

I'm trying to understand fekete problems. There is a variable $s$ and a related concept of 's-energy' [1] [2] [3] [4] that comes up repeatedly when borrowing the concept of potential energy to find ...
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1answer
106 views

find the minimum value of $x^2-6x+9+ \dfrac{64}{x^2}$

Looking for an elegant solution. I can do by brute force, that is finding derivative and double derivative. All Ideas will be appreciated and tried by me.
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1answer
28 views

Simple minimization problem

Suppose we want to execute a program on a processor which can run in three different modes. Each mode can be describe by a pair $(E,\tau)$ where $E$ denotes the energy consumption per cycle (in nJ) ...