1
vote
1answer
130 views

How to solve this optimization problem with equality constraints?

I want to find $\delta_j$ in the following optimization problem. My variables are $\gamma_i$ and $\delta_j$ (all other symbols are known parameters). Assume $i\in\{1,\ldots,9\}$ and ...
3
votes
0answers
39 views

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 ...
0
votes
2answers
63 views

Optimizing elementary symmetric polynomial on the unit sphere

I'd like to optimize $x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$ on the unit 4-sphere. I'm thinking I should do lagrangian optimization, but I'm having trouble solving the resulting ...
1
vote
1answer
45 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
0
votes
1answer
62 views

Lagrange condition and second-order conditions

Given a function to minimize or maximize with equality and/or inequality constraints, I can use Lagrange multiplier and/or KKT to solve such problems. So I understand how it works. My problem is ...
2
votes
0answers
83 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...
0
votes
0answers
49 views

How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
0
votes
1answer
40 views

Uniqueness of Solution to non linear polynomial equations given by lagranges method

When considering Lagrange's method of multipliers for finding maximal solutions to a set of non-linear equations, I have reached a set of 4 equations in 4 real unknowns, $(a,b,c,\lambda)$: ...
1
vote
1answer
52 views

Help to understand the setting up of this Lagrangian

So..I understand up to step 4..but then there are these things I dont get, to start with , it says on (5) that the utility function depended only on the ratios p1/w p2/w ?? why does it say that? ...
1
vote
0answers
72 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...
2
votes
2answers
465 views

How can I solve Lagrange multiplier equation with multi constraints?

This site is really awesome. :) I hope that we can share our ideas through this site! I have an equation as below, $$ min \ \ w^HRw \ \ subject \ \ to \ \ w^HR_aw=J_a, \ w^HR_bw=J_b$$ If there is ...
3
votes
1answer
438 views

Lagrange Multipliers for Function Spaces

For some constant $A > 1$ I am trying to solve the constrained minimization problem minimize $F(u)$ in $C$ subject to $H(u) = 0$. Here $F(u) = \int -u dx$ and $H(u) = \int \sqrt{1 + (u')^2} dx - ...
1
vote
1answer
501 views

Constrained Optimization - Lagrange Multipliers (Example)

Let $f(x, y, z) = xyz$ $h1(x, y, z) = x + y + z − 4$5 and $h2(x, y, z) = 2x − y$. Goal: Minimize $f(x, y, z)$ subject to $h1(x, y, z) = 0$ and $h2(x, y, z) = 0$. First part: Show that every ...
1
vote
1answer
158 views

Lagrange method - non-linear system of equations

I have to compute optimal parametres of truncated cone so that its Volume is fixed (lets say it is 1) and its surface is minimal using Lagrange method These are equations desribing my object: ...