text:
# 5 Exponential cone optimization¶

So far we discussed optimization problems involving the major “polynomial” families of cones: linear, quadratic and power cones. In this chapter we introduce a single new object, namely the three-dimensional exponential cone, together with examples and applications. The exponential cone can be used to model a variety of constraints involving exponentials and logarithms.

## 5.1 Exponential cone¶

The exponential cone is a convex subset of $$\R^3$$ defined as

(5.1)$\begin{split}\begin{array}{rcl} \EXP & = & \left\{(x_1,x_2,x_3)~:~x_1\geq x_2 e^{x_3/x_2},\ x_2>0 \right\}\ \cup \\ & & \left\{(x_1,0,x_3)~:~x_1\geq 0,\ x_3\leq 0 \right\}. \end{array}\end{split}$

Thus the exponential cone is the closure in $$\R^3$$ of the set of points which satisfy

(5.2)$x_1\geq x_2 e^{x_3/x_2},\ x_1,x_2>0.$

When working with logarithms, a convenient reformulation of (5.2) is

(5.3)$x_3 \leq x_2\log(x_1/x_2),\ x_1,x_2>0.$

Alternatively, one can write the same condition as

$x_1/x_2\geq e^{x_3/x_2},\ x_1,x_2>0,$

which immediately shows that $$\EXP$$ is in fact a cone, i.e. $$\alpha x\in \EXP$$ for $$x\in \EXP$$ and $$\alpha\geq 0$$. Convexity of $$\EXP$$ follows from the fact that the Hessian of $$f(x,y)=y\exp(x/y)$$, namely

$\begin{split}D^2(f) = e^{x/y}\left[ \begin{array}{cc} y^{-1} & -xy^{-2} \\ -xy^{-2} & x^2y^{-3} \end{array} \right]\end{split}$

is positive semidefinite for $$y>0$$.

Fig. 5.1 The boundary of the exponential cone $$\EXP$$. The red isolines are graphs of $$x_2 \to x_2\log(x_1/x_2)$$ for fixed $$x_1$$, see (5.3).

## 5.2 Modeling with the exponential cone¶

Extending the conic optimization toolbox with the exponential cone leads to new types of constraint building blocks and new types of representable sets. In this section we list the basic operations available using the exponential cone.

### 5.2.1 Exponential¶

The epigraph $$t\geq e^x$$ is a section of $$\EXP$$:

(5.4)$t\geq e^x \iff (t, 1, x) \in \EXP.$

### 5.2.2 Logarithm¶

Similarly, we can express the hypograph $$t\leq \log{x},\ x\geq 0$$:

(5.5)$t\leq \log{x} \iff (x, 1, t) \in \EXP.$

### 5.2.3 Entropy¶

The entropy function $$H(x) = -x\log{x}$$ can be maximized using the following representation which follows directly from (5.3):

(5.6)$t\leq -x\log{x} \iff t\leq x\log(1/x) \iff (1, x, t) \in \EXP.$

### 5.2.4 Relative entropy¶

The relative entropy or Kullback-Leiber divergence of two probability distributions is defined in terms of the function $$D(x,y)=x\log(x/y)$$. It is convex, and the minimization problem $$t\geq D(x,y)$$ is equivalent to

(5.7)$t\geq D(x,y) \iff -t\leq x\log(y/x) \iff (y, x, -t) \in \EXP.$

Because of this reparametrization the exponential cone is also referred to as the relative entropy cone, leading to a class of problems known as REPs (relative entropy problems). Having the relative entropy function available makes it possible to express epigraphs of other functions appearing in REPs, for instance:

$x \log(1+x/y) = D(x+y,y)+D(y,x+y).$

### 5.2.5 Softplus function¶

In neural networks the function $$f(x)=\log(1+e^x)$$, known as the softplus function, is used as an analytic approximation to the rectifier activation function $$r(x)=x^+=\max(0,x)$$. The softplus function is convex and we can express its epigraph $$t\geq\log(1+e^x)$$ by combining two exponential cones. Note that

$t \geq \log(1+e^x) \iff e^{x-t} + e^{-t} \leq 1$

and therefore $$t\geq\log(1+e^x)$$ is equivalent to the following set of conic constraints:

(5.8)$\begin{split}\begin{array}{rcl} u+v & \leq & 1, \\ (u, 1, x-t) & \in & \EXP, \\ (v, 1, -t) & \in & \EXP. \end{array}\end{split}$

### 5.2.6 Log-sum-exp¶

We can generalize the previous example to a log-sum-exp (logarithm of sum of exponentials) expression

$t \geq \log(e^{x_1}+\cdots+e^{x_n}).$

This is equivalent to the inequality

$e^{x_1-t}+\cdots+e^{x_n-t}\leq 1,$

and so it can be modeled as follows:

(5.9)$\begin{split}\begin{array}{rcl} \sum u_i & \leq & 1, \\ (u_i, 1, x_i-t)& \in & \EXP, \ i=1,\ldots,n. \end{array}\end{split}$

### 5.2.7 Log-sum-inv¶

The following type of bound has applications in capacity optimization for wireless network design:

$t \geq \log\left(\frac{1}{x_1}+\cdots+\frac{1}{x_n}\right),\quad x_i>0.$

Since the logarithm is increasing, we can model this using a log-sum-exp and an exponential as:

$\begin{split}\begin{array}{l} t \geq \log(e^{y_1}+\cdots+e^{y_n}), \\ x_i\geq e^{-y_i},\quad i=1,\ldots,n. \end{array}\end{split}$

### 5.2.8 Arbitrary exponential¶

The inequality

$t \geq a_1^{x_1}a_2^{x_2}\cdots a_n^{x_n},$

where $$a_i$$ are arbitrary positive constants, is of course equivalent to

$t \geq \exp\left(\sum_ix_i\log{a_i}\right)$

and therefore to $$(t, 1, \sum_ix_i\log{a_i})\in\EXP$$.

### 5.2.9 Lambert W-function¶

The Lambert function $$W:\R_+\to \R_+$$ is the unique function satisfying the identity

$W(x)e^{W(x)} = x.$

It is the real branch of a more general function which appears in applications such as diode modeling. The $$W$$ function is concave. Although there is no explicit analytic formula for $$W(x)$$, the hypograph $$\{(x,t)~:~0\leq x,\ 0\leq t\leq W(x)\}$$ has an equivalent description:

$x\geq t e^t = t e^{t^2/t}$

and so it can be modeled with a mix of exponential and quadratic cones (see Sec. 3.1.2 (Rotated quadratic cones)):

(5.10)$\begin{split}\begin{array}{rclr} (x, t, u) & \in & \EXP, & (x\geq t\exp(u/t)), \\ (1/2, u, t) & \in & \Qr, & (u\geq t^2). \end{array}\end{split}$

### 5.2.10 Other simple sets¶

Here are a few more typical sets which can be expressed using the exponential and quadratic cones. The presentations should be self-explanatory; we leave the simple verifications to the reader.

Table 5.1 Sets representable with the exponential cone
Set Conic representation
$$t\geq (\log{x})^2,\, 0< x\leq 1$$ $$(\frac12,t,u)\in\Qr^3,\, (x,1,u)\in\EXP,\, x\leq 1$$
$$t\leq \log\log{x},\, x> 1$$ $$(u,1,t)\in\EXP,\, (x,1,u)\in\EXP$$
$$t\geq (\log{x})^{-1},\, x>1$$ $$(u,t,\sqrt{2})\in\Qr^3,\, (x,1,u)\in\EXP$$
$$t\leq\sqrt{\log{x}},\, x>1$$ $$(\frac12,u,t)\in\Qr^3,\, (x,1,u)\in\EXP$$
$$t\leq\sqrt{x\log{x}},\, x>1$$ $$(x,u,\sqrt{2}t)\in\Qr^3,\, (x,1,u)\in\EXP$$
$$t\leq\log(1-1/x),\, x>1$$ $$(x,u,\sqrt{2})\in\Qr^3,\, (1-u,1,t)\in\EXP$$
$$t\geq\log(1+1/x),\, x>0$$ $$(x+1,u,\sqrt{2})\in\Qr^3,\, (1-u,1,-t)\in\EXP$$

## 5.3 Geometric programming¶

Geometric optimization problems form a family of optimization problems with objective and constraints in special polynomial form. It is a rich class of problems solved by reformulating in logarithmic-exponential form, and thus a major area of applications for the exponential cone $$\EXP$$. Geometric programming is used in circuit design, chemical engineering, mechanical engineering and finance, just to mention a few applications. We refer to [BKVH07] for a survey and extensive bibliography.

### 5.3.1 Definition and basic examples¶

A monomial is a real valued function of the form

(5.11)$f(x_1,\ldots,x_n) = cx_1^{a_1}x_2^{a_2}\cdots x_n^{a_n},$

where the exponents $$a_i$$ are arbitrary real numbers and $$c>0$$. A posynomial (positive polynomial) is a sum of monomials. Thus the difference between a posynomial and a standard notion of a multi-variate polynomial known from algebra or calculus is that (i) posynomials can have arbitrary exponents, not just integers, but (ii) they can only have positive coefficients.

For example, the following functions are monomials (in variables $$x,y,z$$):

(5.12)$xy,\ 2x^{1.5}y^{-1}x^{0.3},\ 3\sqrt{xy/z},\ 1$

and the following are examples of posynomials:

(5.13)$2x+yz,\ 1.5x^3z+5/y,\ (x^2y^2+3z^{-0.3})^4 + 1.$

A geometric program (GP) is an optimization problem of the form

(5.14)$\begin{split}\begin{array}{lll} \minimize & f_0(x) & \\ \st & f_i(x) \leq 1, &i=1,\ldots,m, \\ & x_j>0, &j=1,\ldots,n, \end{array}\end{split}$

where $$f_0,\ldots,f_m$$ are posynomials and $$x=(x_1,\ldots,x_n)$$ is the variable vector.

A geometric program (5.14) can be modeled in exponential conic form by making a substitution

$x_j = e^{y_j},\ j=1,\ldots,n.$

Under this substitution a monomial of the form (5.11) becomes

$ce^{a_1y_1}e^{a_2y_2}\cdots e^{a_ny_n} = \exp(a_\ast^Ty+\log{c})$

for $$a_\ast=(a_1,\ldots,a_n)$$. Consequently, the optimization problem (5.14) takes an equivalent form

(5.15)$\begin{split}\begin{array}{lrcl} \minimize\ t & & & \\ \st & \log(\sum_k\exp(a_{0,k,\ast}^Ty+\log{c_{0,k}}))& \leq & t, \\ & \log(\sum_k\exp(a_{i,k,\ast}^Ty+\log{c_{i,k}}))& \leq & 0,\ i=1,\ldots,m, \end{array}\end{split}$

where $$a_{i,k}\in\R^n$$ and $$c_{i,k}\in\R$$ for all $$i,k$$. These are now log-sum-exp constraints we already discussed in Sec. 5.2.6 (Log-sum-exp). In particular, the problem (5.15) is convex, as opposed to the posynomial formulation (5.14).

Example

We demonstrate this reduction on a simple example. Take the geometric problem

$\begin{split}\begin{array}{lrcl} \minimize & x+y^2z & & \\ \st & 0.1\sqrt{x}+2y^{-1} & \leq & 1, \\ & z^{-1}+yx^{-2} & \leq & 1. \end{array}\end{split}$

By substituting $$x=e^u$$, $$y=e^v$$, $$z=e^w$$ we get

$\begin{split}\begin{array}{lrcl} \minimize\ t & & & \\ \st & \log(e^u+e^{2v+w}) & \leq & t, \\ & \log(e^{0.5u+\log{0.1}}+e^{-v+\log{2}}) & \leq & 0, \\ & \log(e^{-w}+e^{v-2u}) & \leq & 0. \end{array}\end{split}$

and using the log-sum-exp reduction from Sec. 5.2.6 (Log-sum-exp) we write an explicit conic problem:

$\begin{split}\begin{array}{lll} \minimize\ t & \\ \st & (p_1,1,u-t), (q_1,1,2v+w-t) \in \EXP, & p_1+q_1\leq 1,\\ & (p_2,1,0.5u+\log{0.1}), (q_2,1,-v+\log{2}) \in \EXP, & p_2+q_2 \leq 1, \\ & (p_3,1,-w), (q_3,1,v-2u)\in \EXP, & p_3+q_3\leq 1. \end{array}\end{split}$

Solving this problem yields $$(x,y,z)\approx(3.14,2.43,1.32)$$.

### 5.3.2 Generalized geometric models¶

In this section we briefly discuss more general types of constraints which can be modeled with geometric programs.

Monomials

If $$m(x)$$ is a monomial then the constraint $$m(x)=c$$ is equivalent to two posynomial inequalities $$m(x)c^{-1}\leq 1$$ and $$m(x)^{-1}c\leq 1$$, so it can be expressed in the language of geometric programs. In practice it should be added to the model (5.15) as a linear constraint

$\sum_k a_ky_k = \log{c}.$

Monomial inequalities $$m(x)\leq c$$ and $$m(x)\geq c$$ should similarly be modeled as linear inequalities in the $$y_i$$ variables.

In similar vein, if $$f(x)$$ is a posynomial and $$m(x)$$ is a monomial then $$f(x)\leq m(x)$$ is still a posynomial inequality because it can be written as $$f(x)m(x)^{-1}\leq 1$$.

It also means that we can add lower and upper variable bounds: $$0< c_1\leq x\leq c_2$$ is equivalent to $$x^{-1}c_1\leq 1$$ and $$c_2^{-1}x\leq 1$$.

Products and positive powers

Expressions involving products and positive powers (possibly iterated) of posynomials can again be modeled with posynomials. For example, a constraint such as

$((xy^2+z)^{0.3}+y)(1/x+1/y)^{2.2}\leq 1$

can be replaced with

$xy^2+z\leq t,\ t^{0.3}+y\leq u,\ x^{-1}+y^{-1}\leq v,\ uv^{2.2}\leq 1.$

Other transformations and extensions

• If $$f_1,f_2$$ are already expressed by posynomials then $$\max\{f_1(x),f_2(x)\}\leq t$$ is clearly equivalent to $$f_1(x)\leq t$$ and $$f_2(x)\leq t$$. Hence we can add the maximum operator to the list of building blocks for geometric programs.

• If $$f,g$$ are posynomials, $$m$$ is a monomial and we know that $$m(x)\geq g(x)$$ then the constraint $$\frac{f(x)}{m(x)-g(x)}\leq t$$ is equivalent to $$t^{-1}f(x)m(x)^{-1}+g(x)m(x)^{-1}\leq 1$$.

• The objective function of a geometric program (5.14) can be extended to include other terms, for example:

$\minimize\ f_0(x) + \sum_k\log{m_k(x)}$

where $$m_k(x)$$ are monomials. After the change of variables $$x=e^y$$ we get a slightly modified version of (5.15):

$\begin{split}\begin{array}{lrcl} \minimize & t+b^Ty & & \\ \st & \sum_k\exp(a_{0,k,\ast}^Ty+\log{c_{0,k}}) & \leq & t, \\ & \log(\sum_k\exp(a_{i,k,\ast}^Ty+\log{c_{i,k}}))& \leq & 0,\ i=1,\ldots,m, \end{array}\end{split}$

(note the lack of one logarithm) which can still be expressed with exponential cones.

### 5.3.3 Geometric programming case studies¶

Frobenius norm diagonal scaling

Suppose we have a matrix $$M\in\real^{n\times n}$$ and we want to rescale the coordinate system using a diagonal matrix $$D=\Diag(d_1,\ldots,d_n)$$ with $$d_i>0$$. In the new basis the linear transformation given by $$M$$ will now be described by the matrix $$DMD^{-1}=(d_iM_{ij}d_j^{-1})_{i,j}$$. To choose $$D$$ which leads to a “small” rescaling we can for example minimize the Frobenius norm

$\|DMD^{-1}\|_F^2=\sum_{ij} \left((DMD^{-1})_{ij}\right)^2= \sum_{ij} M_{ij}^2d_i^2d_j^{-2}.$

Minimizing the last sum is an example of a geometric program with variables $$d_i$$ (and without constraints).

Maximum likelihood estimation

Geometric programs appear naturally in connection with maximum likelihood estimation of parameters of random distributions. Consider a simple example. Suppose we have two biased coins, with head probabilities $$p$$ and $$2p$$, respectively. We toss both coins and count the total number of heads. Given that, in the long run, we observed $$i$$ heads $$n_i$$ times for $$i=0,1,2$$, estimate the value of $$p$$.

The probability of obtaining the given outcome equals

${n_0+n_1+n_2\choose n_0}{n_1+n_2\choose n_1} (p\cdot 2p)^{n_2}\left(p(1-2p)+2p(1-p)\right)^{n_1}\left((1-p)(1-2p)\right)^{n_0},$

and, up to constant factors, maximizing that expression is equivalent to solving the problem

$\begin{split}\begin{array}{lrcl} \maximize & p^{2n_2+n_1}s^{n_1}q^{n_0}r^{n_0} & & \\ \st & q & \leq & 1-p, \\ & r & \leq & 1-2p, \\ & s & \leq & 3-4p, \\ \end{array}\end{split}$

or, as a geometric problem:

$\begin{split}\begin{array}{lrcl} \minimize & p^{-2n_2-n_1}s^{-n_1}q^{-n_0}r^{-n_0} & & \\ \st & q + p & \leq & 1, \\ & r + 2p & \leq & 1, \\ & \frac{1}{3} s + \frac{4}{3} p & \leq & 1. \\ \end{array}\end{split}$

For example, if $$(n_0,n_1,n_2)=(30,53,16)$$ then the above problem solves with $$p = 0.29$$.

The 26th Vojtěch Jarník International Mathematical Competition, Ostrava 2016. Let $$a,b,c$$ be positive real numbers with $$a+b+c=1$$. Prove that

$\left(\frac{1}{a}+\frac{1}{bc}\right)\left(\frac{1}{b}+\frac{1}{ac}\right)\left(\frac{1}{c}+\frac{1}{ab}\right) \geq 1728$

Using the tricks introduced in Sec. 5.3.2 (Generalized geometric models) we formulate this problem as a geometric program:

$\begin{split}\begin{array}{lrcl} \minimize & pqr & & \\ \st & p^{-1}a^{-1}+p^{-1}b^{-1}c^{-1} & \leq & 1, \\ & q^{-1}b^{-1}+q^{-1}a^{-1}c^{-1} & \leq & 1, \\ & r^{-1}c^{-1}+r^{-1}a^{-1}b^{-1} & \leq & 1, \\ & a+b+c & \leq & 1. \end{array}\end{split}$

Unsurprisingly, the optimal value of this program is $$1728$$, achieved for $$(a,b,c,p,q,r)=\left(\frac13,\frac13,\frac13,12,12,12\right)$$.

Power control and rate allocation in wireless networks

We consider a basic wireless network power control problem. In a wireless network with $$n$$ logical transmitter-receiver pairs if the power output of transmitter $$j$$ is $$p_j$$ then the power received by receiver $$i$$ is $$G_{ij}p_j$$, where $$G_{ij}$$ models path gain and fading effects. If the $$i$$-th receiver’s own noise is $$\sigma_i$$ then the signal-to-interference-plus-noise (SINR) ratio of receiver $$i$$ is given by

(5.16)$s_i = \frac{G_{ii}p_i}{\sigma_i + \sum_{j\neq i} G_{ij}p_j}.$

Maximizing the minimal SINR over all receivers ($$\max \min_i s_i$$), subject to bounded power output of the transmitters, is equivalent to the geometric program with variables $$p_1,\ldots,p_n,t$$:

(5.17)$\begin{split}\begin{array}{lrcr} \minimize & t^{-1} & & \\ \st & p_{\mathrm{min}} \leq p_j\leq p_{\mathrm{max}}, & & j=1,\ldots,n,\\ & t(\sigma_i + \sum_{j\neq i} G_{ij}p_j)G_{ii}^{-1}p_i^{-1} & \leq & 1,\ i=1,\ldots,n. \end{array}\end{split}$

In the low-SNR regime the problem of system rate maximization is approximated by the problem of maximizing $$\sum_{i}\log{s_i}$$, or equivalently minimizing $$\prod_i s_i^{-1}$$. This is a geometric problem with variables $$p_i, s_i$$:

(5.18)$\begin{split}\begin{array}{lrcr} \minimize & s_1^{-1}\cdot \cdots s_n^{-1} & & \\ \st & p_{\mathrm{min}} \leq p_j\leq p_{\mathrm{max}}, & & j=1,\ldots,n,\\ & s_i(\sigma_i + \sum_{j\neq i} G_{ij}p_j)G_{ii}^{-1}p_i^{-1} & \leq & 1,\ i=1,\ldots,n. \end{array}\end{split}$

For more information and examples see [BKVH07].

## 5.4 Exponential cone case studies¶

In this section we introduce some practical optimization problems where the exponential cone comes in handy.

### 5.4.1 Risk parity portfolio¶

Consider a simple version of the Markowitz portfolio optimization problem introduced in Sec. 3.3.3 (Markowitz portfolio optimization), where we simply ask to minimize the risk $$r(x)=\sqrt{x^T\Sigma x}$$ of a fully-invested long-only portfolio:

(5.19)$\begin{split}\begin{array}{lrcl} \minimize & \sqrt{x^T\Sigma x} & & \\ \st & \sum_{i=1}^n x_i & = & 1, \\ & x_i & \geq & 0, \ i=1,\ldots,n, \end{array}\end{split}$

where $$\Sigma$$ is a symmetric positive definite covariance matrix. We can derive from the first-order optimality conditions that the solution to (5.19) satisfies $$\frac{\partial r}{\partial x_i}=\frac{\partial r}{\partial x_j}$$ whenever $$x_i,x_j>0$$, i.e. marginal risk contributions of positively invested assets are equal. In practice this often leads to concentrated portfolios, whereas it would benefit diversification to consider portfolios where all assets have the same total contribution to risk:

(5.20)$x_i\frac{\partial r}{\partial x_i}=x_j\frac{\partial r}{\partial x_j}, \ i,j=1,\ldots,n.$

We call (5.20) the risk parity condition. It indeed models equal risk contribution from all the assets, because as one can easily check $$\frac{\partial r}{\partial x_i} = \frac{(\Sigma x)_i}{\sqrt{x^T\Sigma x}}$$ and

$r(x) = \sum_i x_i\frac{\partial r}{\partial x_i}.$

Risk parity portfolios satisfying condition (5.20) can be found with an auxiliary optimization problem:

(5.21)$\begin{split}\begin{array}{lrl} \minimize & \sqrt{ x^T\Sigma x }- c\sum_i \log{x_i} & \\ \st & x_i > 0, & i=1,\ldots,n, \end{array}\end{split}$

for any $$c>0$$. More precisely, the gradient of the objective function in (5.21) is zero when $$\frac{\partial r}{\partial x_i}=c/x_i$$ for all $$i$$, implying the parity condition (5.20) holds. Since (5.20) is scale-invariant, we can rescale any solution of (5.21) and get a fully-invested risk parity portfolio with $$\sum_i x_i=1$$.

The conic form of problem (5.21) is:

(5.22)$\begin{split}\begin{array}{lrclr} \minimize & t - ce^Ts & & & \\ \st & (t,\Sigma^{1/2}x) & \in & \Q^{n+1}, & (t\geq \sqrt{ x^T\Sigma x }), \\ & (x_i, 1, s_i) & \in & \EXP, & (s_i\leq\log{x_i}). \end{array}\end{split}$

### 5.4.2 Entropy maximization¶

A general entropy maximization problem has the form

(5.23)$\begin{split}\begin{array}{lrcl} \maximize & -\sum_i p_i\log{p_i} & & \\ \st & \sum_i p_i & = & 1, \\ & p_i & \geq & 0, \\ & p & \in & \mathcal{I}, \end{array}\end{split}$

where $$\mathcal{I}$$ defines additional constraints on the probability distribution $$p$$ (these are known as prior information). In the absence of complete information about $$p$$ the maximum entropy principle of Jaynes posits to choose the distribution which maximizes uncertainty, that is entropy, subject to what is known. Practitioners think of the solution to (5.23) as the most random or most conservative of distributions consistent with $$\mathcal{I}$$.

Maximization of the entropy function $$H(x)=-x\log{x}$$ was explained in Sec. 5.2 (Modeling with the exponential cone).

Often one has an a priori distribution $$q$$, and one tries to minimize the distance between $$p$$ and $$q$$, while remaining consistent with $$\mathcal{I}$$. In this case it is standard to minimize the Kullback-Leiber divergence

$\mathcal{D}_{KL}(p||q) = \sum_i p_i \log{p_i/q_i}$

which leads to an optimization problem

(5.24)$\begin{split}\begin{array}{lrcl} \minimize & \sum_i p_i\log{p_i/q_i} & & \\ \st & \sum_i p_i & = & 1, \\ & p_i & \geq & 0, \\ & p & \in & \mathcal{I}. \end{array}\end{split}$

### 5.4.3 Hitting time of a linear system¶

Consider a linear dynamical system

(5.25)$\mathbf{x}'(t) = A \mathbf{x}(t)$

where we assume for simplicity that $$A=\Diag(a_1,\ldots,a_n)$$ with $$a_i<0$$ with initial condition $$\mathbf{x}(0)_i=x_i$$. The resulting dynamical system $$\mathbf{x}(t)=\mathbf{x}(0)\exp(At)$$ converges to $$0$$ and one can ask, for instance, for the time it takes to approach the limit up to distance $$\varepsilon$$. The resulting optimization problem is

$\begin{split}\begin{array}{lrcl} \minimize & t & & \\ \st & \sqrt{\sum_i \left(x_i\exp(a_it)\right)^2} & \leq & \varepsilon, \end{array}\end{split}$

with the following conic form, where the variables are $$t, q_1, \ldots,q_n$$:

$\begin{split}\begin{array}{lrcl} \minimize & t & & \\ \st & (\varepsilon,x_1q_1,\ldots,x_nq_n) & \in & \Q^{n+1},\\ & (q_i, 1, a_it) & \in & \EXP, \ i=1,\ldots,n. \end{array}\end{split}$

See Fig. 5.2 for an example. Other criteria for the target set of the trajectories are also possible. For example, polyhedral constraints

$c^T\mathbf{x}\leq d, \ c\in \real_+^n, d \in \real_+$

are also expressible in exponential conic form for starting points $$\mathbf{x}(0)\in\real_+^n$$, since they correspond to log-sum-exp constraints of the form

$\log\left(\sum_i \exp(a_i\mathbf{x}_i + \log(c_ix_i))\right) \leq \log{d}.$

For a robust version involving uncertainty on $$A$$ and $$x(0)$$ see [CS14].

Fig. 5.2 With $$A=\Diag(-0.3, -0.06)$$ and starting point $$x(0)=(2.2, 1.3)$$ the trajectory reaches distance $$\varepsilon=0.5$$ from origin at time $$t\approx 15.936$$.

### 5.4.4 Logistic regression¶

Logistic regression is a technique of training a binary classifier. We are given a training set of examples $$x_1, \ldots, x_n \in\real^d$$ together with labels $$y_i\in\{0,1\}$$. The goal is to train a classifier capable of assigning new data points to either class 0 or 1. Specifically, the labels should be assigned by computing

(5.26)$h_\theta(x)=\frac{1}{1+\exp(-\theta^Tx)}$

and choosing label $$y=0$$ if $$h_\theta(x)<\frac12$$ and $$y=1$$ for $$h_\theta(x)\geq \frac12$$. Here $$h_\theta(x)$$ is interpreted as the probability that $$x$$ belongs to class 1. The optimal parameter vector $$\theta$$ should be learned from the training set, so as to maximize the likelihood function:

$\prod_i h_\theta(x_i)^{y_i}(1-h_\theta(x_i))^{1-y_i}.$

By taking logarithms and adding regularization with respect to $$\theta$$ we reach an unconstrained optimization problem

(5.27)$\minimize_{\theta\in\real^d}\ \lambda\|\theta\|_2 + \sum_i -y_i\log(h_\theta(x_i))-(1-y_i)\log(1-h_\theta(x_i)).$

Problem (5.27) is convex, and can be more explicitly written as

$\begin{split}\begin{array}{lrlll} \minimize & \sum_i t_i +\lambda r & & & \\ \st & t_i & \geq - \log(h_\theta(x)) &= \log(1+\exp(-\theta^Tx_i)) &\ \mathrm{if}\ y_i=1, \\ & t_i & \geq - \log(1-h_\theta(x)) &= \log(1+\exp(\theta^Tx_i)) &\ \mathrm{if}\ y_i=0, \\ & r & \geq \|\theta\|_2, & & \end{array}\end{split}$

involving softplus type constraints (see Sec. 5.2 (Modeling with the exponential cone)) and a quadratic cone. See Fig. 5.3 for an example.

Fig. 5.3 Logistic regression example with none, medium and strong regularization (small, medium, large $$\lambda$$). The two-dimensional dataset was converted into a feature vector $$x\in\real^{28}$$ using monomial coordinates of degrees at most 6. Without regularization we get obvious overfitting.


text:
# What is the overall maximum value of the function f(x)=(18x)/(4+x^2) on the interval (0,4)?

$x = 2$
$y = 4.5$

format: plain


text:
# Integration involving Roots

1. Aug 10, 2006

### p4nda

I am stuck half way in solving this problem (the square root nominator confuses me) :
http://img235.imageshack.us/img235/8459/1mi1.jpg [Broken]

and I cannot get it to match the answer given on the back of the textbook:

Please teach me how to solve this problem. Thanks. :)

Last edited by a moderator: May 2, 2017
2. Aug 10, 2006

### d_leet

First making a substitution would help, let u = x1/2. Then you can factor the denominator and possibly break the integrand into partial fractions.

3. Aug 10, 2006

### WigneRacah

The right substitution is $$u=x^{\frac{1}{6}}$$.

Thus we have:

$$\int \frac{x^{1/2}}{1 + x^{1/3}} dx = 6 \int \frac{u^8}{1+u^2} du = 6 \int (u^6 - u^4 + u^2 -1 + \frac{1}{1+u^2}) du$$.

4. Aug 10, 2006

### d_leet

Whoops, you're right about that, I missed that it was a cube root in the denominator.

5. Aug 10, 2006

### VietDao29

Only to explain a little bit further. Of course, We then cannot use Partial Fraction if there are roots (square roots, or whatever) in our expression. So, the aim is that, we should make all the square roots, or cube roots in the problem disappear. And so, we choose $$u = \sqrt[6]{x}$$, or $$u ^ 6 = x$$.
We have:
$$u ^ 6 = x \Rightarrow 6 u ^ 5 du = dx$$
So: $$\sqrt{x} = \sqrt{u ^ 6} = u ^ 3$$, and $$\sqrt[3]{x} = \sqrt[3]{u ^ 6} = u ^ 2$$.
All the roots now have disappeared.
We then can use Partial Fraction, and integrate it. :)


text:
# Event-related potential

A waveform showing several ERP components, including the N100 (labeled N1) and P300 (labeled P3). Note that the ERP is plotted with negative voltages upward, a common, but not universal, practice in ERP research

An event-related potential (ERP) is the measured brain response that is the direct result of a specific sensory, cognitive, or motor event.[1] More formally, it is any stereotyped electrophysiological response to a stimulus. The study of the brain in this way provides a noninvasive means of evaluating brain functioning.

ERPs are measured by means of electroencephalography (EEG). The magnetoencephalography (MEG) equivalent of ERP is the ERF, or event-related field.[2] Evoked potentials and induced potentials are subtypes of ERPs.

## History

With the discovery of the electroencephalogram (EEG) in 1924, Hans Berger revealed that one could measure the electrical activity of the human brain by placing electrodes on the scalp and amplifying the signal. Changes in voltage can then be plotted over a period of time. He observed that the voltages could be influenced by external events that stimulated the senses. The EEG proved to be a useful source in recording brain activity over the ensuing decades. However, it tended to be very difficult to assess the highly specific neural process that are the focus of cognitive neuroscience because using pure EEG data made it difficult to isolate individual neurocognitive processes. Event-related potentials (ERPs) offered a more sophisticated method of extracting more specific sensory, cognitive, and motor events by using simple averaging techniques. In 1935–1936, Pauline and Hallowell Davis recorded the first known ERPs on awake humans and their findings were published a few years later, in 1939. Due to World War II not much research was conducted in the 1940s, but research focusing on sensory issues picked back up again in the 1950s. In 1964, research by Grey Walter and colleagues began the modern era of ERP component discoveries when they reported the first cognitive ERP component, called the contingent negative variation (CNV).[3] Sutton, Braren, and Zubin (1965) made another advancement with the discovery of the P3 component.[4] Over the next fifteen years, ERP component research became increasingly popular. The 1980s, with the introduction of inexpensive computers, opened up a new door for cognitive neuroscience research. Currently, ERP is one of the most widely used methods in cognitive neuroscience research to study the physiological correlates of sensory, perceptual and cognitive activity associated with processing information.[5]

## Calculation

ERPs can be reliably measured using electroencephalography (EEG), a procedure that measures electrical activity of the brain over time using electrodes placed on the scalp. The EEG reflects thousands of simultaneously ongoing brain processes. This means that the brain response to a single stimulus or event of interest is not usually visible in the EEG recording of a single trial. To see the brain's response to a stimulus, the experimenter must conduct many trials and average the results together, causing random brain activity to be averaged out and the relevant waveform to remain, called the ERP.[6]

The random (background) brain activity together with other bio-signals (e.g., EOG, EMG, EKG) and electromagnetic interference (e.g., line noise, fluorescent lamps) constitute the noise contribution to the recorded ERP. This noise obscures the signal of interest, which is the sequence of underlying ERPs under study. From an engineering point of view it is possible to define the signal-to-noise ratio (SNR) of the recorded ERPs. Averaging increases the SNR of the recorded ERPs making them discernible and allowing for their interpretation. This has a simple mathematical explanation provided that some simplifying assumptions are made. These assumptions are:

1. The signal of interest is made of a sequence of event-locked ERPs with invariable latency and shape
2. The noise can be approximated by a zero-mean Gaussian random process of variance ${\displaystyle \sigma ^{2}}$ which is uncorrelated between trials and not time-locked to the event (this assumption can be easily violated, for example in the case of a subject doing little tongue movements while mentally counting the targets in an experiment).

Having defined ${\displaystyle k}$, the trial number, and ${\displaystyle t}$, the time elapsed after the ${\displaystyle k}$th event, each recorded trial can be written as ${\displaystyle x(t,k)=s(t)+n(t,k)}$ where ${\displaystyle s(t)}$ is the signal and ${\displaystyle n(t,k)}$ is the noise (Note that, under the assumptions above, the signal does not depend on the specific trial while the noise does).

The average of ${\displaystyle N}$ trials is

${\displaystyle {\bar {x}}(t)={\frac {1}{N}}\sum _{k=1}^{N}x(t,k)=s(t)+{\frac {1}{N}}\sum _{k=1}^{N}n(t,k)}$ .

The expected value of ${\displaystyle {\bar {x}}(t)}$ is (as hoped) the signal itself, ${\displaystyle \operatorname {E} [{\bar {x}}(t)]=s(t)}$.

Its variance is

${\displaystyle \operatorname {Var} [{\bar {x}}(t)]=\operatorname {E} \left[\left({\bar {x}}(t)-\operatorname {E} [{\bar {x}}(t)]\right)^{2}\right]={\frac {1}{N^{2}}}\operatorname {E} \left[\left(\sum _{k=1}^{N}n(t,k)\right)^{2}\right]={\frac {1}{N^{2}}}\sum _{k=1}^{N}\operatorname {E} \left[n(t,k)^{2}\right]={\frac {\sigma ^{2}}{N}}}$.

For this reason the noise amplitude of the average of ${\displaystyle N}$ trials is expected to deviate from the mean (which is ${\displaystyle s(t)}$) by less or equal than ${\displaystyle \sigma /{\sqrt {N}}}$ in 68% of the cases. In particular, the deviation wherein 68% of the noise amplitudes lie is ${\displaystyle 1/{\sqrt {N}}}$ times that of a single trial. A larger deviation of ${\displaystyle 2\sigma /{\sqrt {N}}}$ can already be expected to encompass 95% of all noise amplitudes.

Wide amplitude noise (such as eye blinks or movement artifacts) are often several orders of magnitude larger than the underlying ERPs. Therefore, trials containing such artifacts should be removed before averaging. Artifact rejection can be performed manually by visual inspection or using an automated procedure based on predefined fixed thresholds (limiting the maximum EEG amplitude or slope) or on time-varying thresholds derived from the statistics of the set of trials.[citation needed]

## Nomenclature

ERP waveforms consist of a series of positive and negative voltage deflections, which are related to a set of underlying components.[7] Though some ERP components are referred to with acronyms (e.g., contingent negative variation – CNV, error-related negativity – ERN), most components are referred to by a letter (N/P) indicating polarity (negative/positive), followed by a number indicating either the latency in milliseconds or the component's ordinal position in the waveform. For instance, a negative-going peak that is the first substantial peak in the waveform and often occurs about 100 milliseconds after a stimulus is presented is often called the N100 (indicating its latency is 100 ms after the stimulus and that it is negative) or N1 (indicating that it is the first peak and is negative); it is often followed by a positive peak, usually called the P200 or P2. The stated latencies for ERP components are often quite variable, particularly so for the later components that are related to the cognitive processing of the stimulus. For example, the P300 component may exhibit a peak anywhere between 250 ms – 700 ms.

### Relative to behavioral measures

Compared with behavioral procedures, ERPs provide a continuous measure of processing between a stimulus and a response, making it possible to determine which stage(s) are being affected by a specific experimental manipulation. Another advantage over behavioral measures is that they can provide a measure of processing of stimuli even when there is no behavioral change. However, because of the significantly small size of an ERP, it usually takes a large number of trials to accurately measure it correctly.[8]

### Relative to other neurophysiological measures

#### Invasiveness

Unlike microelectrodes, which require an electrode to be inserted into the brain, and PET scans that expose humans to radiation, ERPs use EEG, a non-invasive procedure.

#### Spatial and temporal resolution

ERPs provide excellent temporal resolution—as the speed of ERP recording is only constrained by the sampling rate that the recording equipment can feasibly support, whereas hemodynamic measures (such as fMRI, PET, and fNIRS) are inherently limited by the slow speed of the BOLD response. The spatial resolution of an ERP, however, is much poorer than that of hemodynamic methods—in fact, the location of ERP sources is an inverse problem that cannot be exactly solved, only estimated. Thus, ERPs are well suited to research questions about the speed of neural activity, and are less well suited to research questions about the location of such activity.[1]

### Cost

ERP research is much cheaper to do than other imaging techniques such as fMRI, PET, and MEG. This is because purchasing and maintaining an EEG system is less expensive than the other systems.

## Clinical

Physicians and neurologists will sometimes use a flashing visual checkerboard stimulus to test for any damage or trauma in the visual system. In a healthy person, this stimulus will elicit a strong response over the primary visual cortex located in the occipital lobe, in the back of the brain.

ERP component abnormalities in clinical research have been shown in neurological conditions such as:

## Research

ERPs are used extensively in neuroscience, cognitive psychology, cognitive science, and psycho-physiological research. Experimental psychologists and neuroscientists have discovered many different stimuli that elicit reliable ERPs from participants. The timing of these responses is thought to provide a measure of the timing of the brain's communication or timing of information processing. For example, in the checkerboard paradigm described above, healthy participants' first response of the visual cortex is around 50–70 ms. This would seem to indicate that this is the amount of time it takes for the transduced visual stimulus to reach the cortex after light first enters the eye. Alternatively, the P300 response occurs at around 300ms in the oddball paradigm, for example, regardless of the type of stimulus presented: visual, tactile, auditory, olfactory, gustatory, etc. Because of this general invariance with regard to stimulus type, the P300 component is understood to reflect a higher cognitive response to unexpected and/or cognitively salient stimuli. The P300 response has also been studied in the context of information and memory detection.[21] In addition, there are studies on abnormalities of P300 in depression. Depressed patients tend to have a reduced P200 and P300 amplitude and a prolonged P300 latency.[19]

Due to the consistency of the P300 response to novel stimuli, a brain-computer interface can be constructed which relies on it. By arranging many signals in a grid, randomly flashing the rows of the grid as in the previous paradigm, and observing the P300 responses of a subject staring at the grid, the subject may communicate which stimulus he is looking at, and thus slowly "type" words.[22]

Another area of research in the field of ERP lies in the efference copy. This predictive mechanism plays a central role in for example human verbalization.[23][24] Efference copies, however, do not only occur with spoken words, but also with inner language - i.e. the quiet production of words - which has also been proven by event-related potentials.[25]

Other ERPs used frequently in research, especially neurolinguistics research, include the ELAN, the N400, and the P600/SPS. The analysis of ERP data is also increasingly supported by machine learning algorithms.[26][27]

### Number of Trials

A common issue in ERP studies is whether the observed data have a sufficient number of trials to support statistical analysis. The background noise in any ERP for any individual can vary. Therefore simply characterizing the number of ERP trials needed for a robust component response is inadequate. Therefore, ERP researchers can use metrics like the standardized measurement error (SME) to justify the examination of between-condition or between-group differences [28] or estimates of internal consistency to justify the examination of individual differences.[29][30]

## References

1. ^ a b Luck SJ (2005). An Introduction to the Event-Related Potential Technique. The MIT Press. ISBN 978-0-262-12277-1.[page needed]
2. ^ Brown CM, Hagoort P (1999). "The cognitive neuroscience of language". In Brown CM, Hagoort P (eds.). The Neurocognition of Language. New York: Oxford University Press. p. 6.
3. ^ Walter WG, Cooper R, Aldridge VJ, Mccallum WC, Winter AL (July 1964). "Contingent Negative Variation: An Electric Sign of Sensori-Motor Association and Expectancy in the Human Brain". Nature. 203 (4943): 380–4. Bibcode:1964Natur.203..380W. doi:10.1038/203380a0. PMID 14197376. S2CID 26808780.
4. ^ Sutton S, Braren M, Zubin J, John ER (November 1965). "Evoked-potential correlates of stimulus uncertainty". Science. 150 (3700): 1187–8. Bibcode:1965Sci...150.1187S. doi:10.1126/science.150.3700.1187. PMID 5852977. S2CID 39822117.
5. ^ Handy, T. C. (2005). Event Related Potentials: A Methods Handbook. Cambridge, Massachusetts: Bradford/MIT Press.[page needed]
6. ^ Coles MG, Rugg MD (1995). "Event-related brain potentials: An introduction". In Rugg MD, Coles MG (eds.). Electrophysiology of mind: Event-related brain potentials and cognition. Oxford psychology series, No. 25. New York: Oxford University Press. pp. 1–26.
7. ^ Luck SJ, Kappenman ES, eds. (2012). The Oxford Handbook of Event-Related Potential Components. Oxford University Press. p. 664. ISBN 9780195374148.
8. ^ Luck S (2005). "Comparison with Behavioral Measures". An Introduction to the Event-Related Potential Technique. MIT Press. pp. 21–23.
9. ^ Johnstone SJ, Barry RJ, Clarke AR (April 2013). "Ten years on: a follow-up review of ERP research in attention-deficit/hyperactivity disorder". Clinical Neurophysiology. 124 (4): 644–57. doi:10.1016/j.clinph.2012.09.006. PMID 23063669. S2CID 13867965.
10. ^ Barry RJ, Johnstone SJ, Clarke AR (February 2003). "A review of electrophysiology in attention-deficit/hyperactivity disorder: II. Event-related potentials". Clinical Neurophysiology. 114 (2): 184–98. doi:10.1016/S1388-2457(02)00363-2. PMID 12559225. S2CID 9239459.
11. ^ Boutros N, Torello MW, Burns EM, Wu SS, Nasrallah HA (June 1995). "Evoked potentials in subjects at risk for Alzheimer's disease". Psychiatry Research. 57 (1): 57–63. doi:10.1016/0165-1781(95)02597-P. PMID 7568559. S2CID 17010156.
12. ^ Prabhakar S, Syal P, Srivastava T (September 2000). "P300 in newly diagnosed non-dementing Parkinson's disease: effect of dopaminergic drugs". Neurology India. 48 (3): 239–42. PMID 11025627.
13. ^ Boose MA, Cranford JL (January 1996). "Auditory event-related potentials in multiple sclerosis". The American Journal of Otology. 17 (1): 165–70. PMID 8694124.
14. ^ Duncan CC, Kosmidis MH, Mirsky AF (January 2003). "Event-related potential assessment of information processing after closed head injury". Psychophysiology. 40 (1): 45–59. doi:10.1111/1469-8986.00006. PMID 12751803.
15. ^ D'Arcy RC, Marchand Y, Eskes GA, Harrison ER, Phillips SJ, Major A, Connolly JF (April 2003). "Electrophysiological assessment of language function following stroke". Clinical Neurophysiology. 114 (4): 662–72. doi:10.1016/S1388-2457(03)00007-5. PMID 12686275. S2CID 27955719.
16. ^ Hanna GL, Carrasco M, Harbin SM, Nienhuis JK, LaRosa CE, Chen P, et al. (September 2012). "Error-related negativity and tic history in pediatric obsessive-compulsive disorder". Journal of the American Academy of Child and Adolescent Psychiatry. 51 (9): 902–10. doi:10.1016/j.jaac.2012.06.019. PMC 3427894. PMID 22917203.
17. ^ Ford JM, Palzes VA, Roach BJ, Mathalon DH (July 2014). "Did I do that? Abnormal predictive processes in schizophrenia when button pressing to deliver a tone". Schizophrenia Bulletin. 40 (4): 804–12. doi:10.1093/schbul/sbt072. PMC 4059422. PMID 23754836.
18. ^ Clayson PE, Wynn JK, Infantolino ZP, Hajcak G, Green MF, Horan WP (November 2019). "Reward processing in certain versus uncertain contexts in schizophrenia: An event-related potential (ERP) study". Journal of Abnormal Psychology. 128 (8): 867–880. doi:10.1037/abn0000469. PMC 6822386. PMID 31657597.
19. ^ a b Zhou L, Wang G, Nan C, Wang H, Liu Z, Bai H (January 2019). "Abnormalities in P300 components in depression: an ERP-sLORETA study". Nordic Journal of Psychiatry. 73 (1): 1–8. doi:10.1080/08039488.2018.1478991. PMID 30636465. S2CID 58664019.
20. ^ Casanova MF, Sokhadze EM, Casanova EL, Li X (October 2020). "Transcranial Magnetic Stimulation in Autism Spectrum Disorders: Neuropathological Underpinnings and Clinical Correlations". Seminars in Pediatric Neurology. 35: 100832. doi:10.1016/j.spen.2020.100832. PMC 7477302. PMID 32892959.
21. ^ McCormick B (2006). "Your Thoughts May Deceive You: The Constitutional Implications of Brain Fingerprinting Technology and How It May Be Used to Secure Our Skies". Law & Psychology Review. 30: 171–84.
22. ^ Farwell LA, Donchin E (December 1988). "Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials". Electroencephalography and Clinical Neurophysiology. 70 (6): 510–23. doi:10.1016/0013-4694(88)90149-6. PMID 2461285.
23. ^ Roach BJ, Ford JM, Biagianti B, Hamilton HK, Ramsay IS, Fisher M, et al. (November 2019). "Efference copy/corollary discharge function and targeted cognitive training in patients with schizophrenia". International Journal of Psychophysiology. 145: 91–98. doi:10.1016/j.ijpsycho.2018.12.015. PMC 6616012. PMID 30599145.
24. ^ Brumberg JS, Pitt KM (July 2019). "Motor-Induced Suppression of the N100 Event-Related Potential During Motor Imagery Control of a Speech Synthesizer Brain-Computer Interface". Journal of Speech, Language, and Hearing Research. 62 (7): 2133–2140. doi:10.1044/2019_JSLHR-S-MSC18-18-0198. PMC 6808362. PMID 31306609.
25. ^ Whitford TJ, Jack BN, Pearson D, Griffiths O, Luque D, Harris AW, et al. (December 2017). "Neurophysiological evidence of efference copies to inner speech". eLife. 6. doi:10.7554/eLife.28197. PMC 5714499. PMID 29199947.
26. ^ Mueller A, Candrian G, Kropotov JD, Ponomarev VA, Baschera GM (June 2010). "Classification of ADHD patients on the basis of independent ERP components using a machine learning system". Nonlinear Biomedical Physics. 4 (Suppl 1): S1. doi:10.1186/1753-4631-4-S1-S1. PMC 2880795. PMID 20522259.
27. ^ Frick J, Rieg T, Buettner R (2021). Detection of schizophrenia: a machine learning algorithm for potential early detection and prevention based on event-related potentials. Proceedings of the 54th Hawaii International Conference on System Sciences. doi:10.24251/HICSS.2021.460.
28. ^ Luck SJ, Stewart AX, Simmons AM, Rhemtulla M (June 2021). "Standardized measurement error: A universal metric of data quality for averaged event-related potentials". Psychophysiology. 58 (6): e13793. doi:10.1111/psyp.13793. PMC 8169536. PMID 33782996.
29. ^ Clayson PE, Miller GA (January 2017). "Psychometric considerations in the measurement of event-related brain potentials: Guidelines for measurement and reporting". International Journal of Psychophysiology. 111: 57–67. doi:10.1016/j.ijpsycho.2016.09.005. PMID 27619493.
30. ^ Clayson PE, Brush CJ, Hajcak G (July 2021). "Data quality and reliability metrics for event-related potentials (ERPs): The utility of subject-level reliability". International Journal of Psychophysiology. 165: 121–136. doi:10.1016/j.ijpsycho.2021.04.004. PMID 33901510.


text:
Thread: Integral - Multiplying by a form of one

1. Integral - Multiplying by a form of one

Apparently, I can multiply this by a fraction equal to one that contains some trig functions.

$
\int {\frac{1}
{{1 + \sin x}}dx}
$

I can't seem to find this "trick" multiplier simplify this. I've tried cosx/cosx , cscx/cscx, cotx/cotx, etc.

2. Originally Posted by RedBarchetta
Apparently, I can multiply this by a fraction equal to one that contains some trig functions.

$
\int {\frac{1}
{{1 + \sin x}}dx}
$

I can't seem to find this "trick" multiplier simplify this. I've tried cosx/cosx , cscx/cscx, cotx/cotx, etc.
$\frac{1}{1+\sin x} \cdot \frac{1-\sin x}{1-\sin x} = \frac{1-\sin x}{\cos^2 x} = \sec^2 x - \frac{\sin x}{\cos^2 x}$

3. Originally Posted by ThePerfectHacker
$\frac{1}{1+\sin x} \cdot \frac{1-\sin x}{1-\sin x} = \frac{1-\sin x}{\cos^2 x} = \sec^2 x - \frac{\sin x}{\cos^2 x}$
Thanks. Is there any way to learn how to recognize these patterns better or a certain technique that you use?

4. Originally Posted by RedBarchetta
Thanks. Is there any way to learn how to recognize these patterns better or a certain technique that you use?
A lot of comes from experience.
The more problems you do the more ideas you get to how to solve problems.

5. Here, you have 1+sin(x)

Try to find identities with 1 and sin(x). The first that comes in mind is cos²(x)+sin²(x)=1 ---> 1-sin²(x)=cos²(x)=(1-sin(x))(1+sin(x))

When you're dealing with cot, or tan, it can be interesting to first multiply by cos or sin, regarding the situation.

When you're dealing with a sum of cos or a sum of sin, recall the identities of addition sin(a+b), cos(a+b)... cos(a)+cos(b)=.... sin(a)+sin(b)=....

But it's true, like TPH says, a lot comes with experience.