p***o
发帖数: 1252 | 1 你把那个约束通分化简成 (1/x+1/y)(1/a+1/b) <= 1，手算估计都能得到结果。 |
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i******t
发帖数: 370 | 2 cvx可以解geometric programming，不过我没用过。
实在不行自己gradient descent也可以，反正就4个variables |
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w*******i
发帖数: 525 | 3 这个可以转成geometric programming？如何？ |
|
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p**o
发帖数: 3409 | 5 我把我的推导贴一下，还烦高人帮忙看看问题在哪儿～～
\min f(x) = \min \sum_i p(x_i) = \min \sum_i x_i^{1/2}
s.t.
h(x) = \sum_i x_i - C_0 = 0
为减少乘子数量，0\leq xi\leq Ci看作f(x)的定义域而不是优化约束
1. 先找p(x_i)=x_i^{1/2}的convex conjugate:
p^*(y_i) = \sup_{0\leq xi\leq Ci} (y_i x_i - p(x_i))
y_i x_i - p(x_i)对x_i二阶导大于0，上届在两端取到，所以
p^*(y_i) =
C_i y_i - p(C_i), if y_i > p(C_i)/C_i
0, otherwise
2. f(x) = \sum_i p(x_i)各分量独立，所以f(x)的convex conjugate为
f^*(y)
= \sum_i p^*(y_i)
= \sum_{i:y_i>p(C_i)/C_i} (C_i y_i - p(C_i))
3. Lagrangian of f(x)
L(x,v) |
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g*********n
发帖数: 119 | 6 来自主题: Mathematics版 - 想自学优化 s. boyd, convex programming这本书找不着啊, 是convex optimization? |
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D**u
发帖数: 204 | 7 Here is a "小" but interesting one.
It was conjectured that "polyhedra are rigid" many years ago by
names like Euler. The convex case was long known to be true,
but not until the 70's a counterexample was found
for a non-convex one.
Here "polyhedra" are formed by rigid faces, but the angles
between connecting faces are allowed to change.
See:
http://en.wikipedia.org/wiki/Flexible_polyhedron |
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n***p
发帖数: 7668 | 8 Hope this problem is not your homework. I thought
about it and give the following predictions. Nothing
is guaranteed to be true though.
Apparently what you need to do is to minimize E over
the admissible set
S={u=(u1,u2)\in H^1(D, R^2): u2=0 on the boundary,
u1 u1在上下边非负，在左右边非正 }.
Now S is a closed, convex subset of H^1(D, R^2). Using
relatively standard arguments in calculus of variations,
There exists a minimizer for E on S. But since E is not
a convex functional, th... 阅读全帖 |
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u*****l
发帖数: 444 | 9 我开会碰到一个低年级PHD,是我见过最牛B的PHD, 本科的时候发了一堆paper了,包括
Ann of Math. 这人刚从普林毕业,在斯坦福第一年还是第二年,刚做了Hilbert--Smith
conjecture for three-manifolds.做的领域也相当广,代数几何,拓扑,概率都做. 个人
感觉这个人比同时期的TAO要强.
The Hilbert--Smith conjecture for three-manifolds. Submitted. (April 2012).
The link concordance invariant from Lee homology. Algebr. Geom. Topol. 12 (
2012), no. 2, 1081--1098. Published Version. MR2928905.
(with János Kollár). Algebraic varieties with semialgebraic universal
cover. J. Topol. 5 (2012), no. 1, 199--212. Publi... 阅读全帖 |
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E*****T
发帖数: 1193 | 10 I guess the difficulty is that the space of continous function with sup norm
is not reflexive. You can try to prove it with additional condition: F
closed, convex (maybe also bounded?) I don't know if this is right.
But if you let F be a closed, convex (bounded?) subset of some Sobolev space
which can be embedded into C^0, then a minimizer should exists, since
Sobolev space is reflexive. |
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c**a
发帖数: 316 | 11 Dual cones satisfy several properties, such as:
* ...
* ...
* ...
* ...
* K** is the closure of the convex hull of K. (Hence, if K is convex and
closed, K** = K).
The content in the parenthesis was by the original authors, i.e. Boyd. |
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x********i
发帖数: 905 | 12 http://iccm.mcm.ac.cn/dct/page/1
Invited Lectures
Group 1
Fan Qin: Cluster algebras and monoidal categorification
Fang Li: Positivity of acyclic sign-skew-symmetric cluster algebras via
unfolding method and some related topics
Cheng-Chiang Tsai: An attempt for affine Springer theory
Li Cai: The Gross-Zagier formula: arithmetic applications
Ming-Hsuan Kang: Geometric zeta functions on reductive groups over non-
archimedean local fields
Huanchen Bao: Canonical bases arising... 阅读全帖 |
|
l******f
发帖数: 568 | 13 let f(x) be a convex function (e^x in this context) and l(x)=a+b*x be a
tangent line to f(x) at the point f(E(x)) for some a and b.
by convexity f(x)>=a+b*x=l(x)
Take expectation on both sides, E(f(x))>=E(a+b*x)=a+b*E(x)=l(E(x))=f(E(x)) |
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T*******t
发帖数: 9274 | 14
in
I'd suggest you asking some senior guys in your group to get some idea.
I cannot believe any tree models in production level are not able to
price the swap you mentioned.
For your swap, no need to get tree involved. You even don't need
convexity adjustment if the trade is not super long or highly leveraged as
the adjustment should be small.
If you know how to calculate the accurate prices of EuroDollar Futures,
you should have no problem to get the convexity adjustment for your swap. |
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y*w
发帖数: 238 | 15
这个函数不是convex的
只能在f（y）=sum [（M_i-y）^2 ] 对于y递增的区域是convex的
你plot个图看看就知道了，比如 (exp(-x)-1)^2在【0，2】之间
不过貌似可以证明是伪凸还不知道是准凸，还是可以保证有global最优的 |
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b***k
发帖数: 2673 | 16 ☆─────────────────────────────────────☆
keku2000 (呵呵) 于 (Wed May 28 11:39:39 2008) 提到:
除了按照定义硬算
还有什么快的办法么
☆─────────────────────────────────────☆
hucat307 (阿猫) 于 (Wed May 28 11:55:44 2008) 提到:
the dynamic has dt term.
or use the fact x^3 is a convex function
☆─────────────────────────────────────☆
qos (CTer) 于 (Wed May 28 16:15:22 2008) 提到:
x^3 is not convex.
One can use Ito's formula, or one can check directly by the definition of
martingale.
☆─────────────────────────────────── |
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w*****e
发帖数: 197 | 17 I agree that these option tricks are boresome.
But PingMianJiHe definitely helps even if you
can do everything analytically. It is like
in math, a problem can be solved with a very
sophisticated method. But an elementary method
will be highly appreciated. It just shows how
much insight you have into the problem.
Put it another way, you can say BS formula tells
you options are convex. Or you can use a simple
no-arbitrage argument to show that options should
be convex. The latter is so much better |
|
l****o
发帖数: 2909 | 18 1. Mishra SK, Wang SY, Lai KK, Explicitly B-preinvex fuzzy mappings,
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS , 83 (1): 39-47 JAN 2006.
2. Lean Yu, Shouyang Wang, K.K. Lai, An Integrated Data Preparation Scheme
for Neural Network Data Analysis, IEEE Transactions on Knowledge and Data
Engineering, 18, pp.1-13, 2006.
3. Kin Keung Lai, Lean Yu, Shouyang Wang, Multi-agent Web Text Mining on the
Grid for Enterprise Decision Support, Lecture Notes in Computer Science,
3842, pp. 540 - 544, 2006.
4... 阅读全帖 |
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f********y
发帖数: 278 | 19 Steven E. Shreve第一册111-113页，主要是证明为什么american call不需要在
expiration day之前执行。我看他在证明的时候只用了两个条件：a,call payment的曲
线是convex，b,Jensen's inequality.但是american put也是满足这两个条件的，尽管
他说公式4.5.6对american put不适用，但是4.5.6是从payment曲线的convex特性而来
得阿？
谢谢指点! |
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g******n
发帖数: 7 | 20 Gamma is the second order derivative of call option price w.r.t the
underlying price.
Second order derivative describs convexity.
Simple call option is always convex, so gamma is positive. |
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h*y
发帖数: 1289 | 21 Nice discussion. I also have a question. My understanding is that the gamma
is similar to the convexity in Fixed Income, which is positive for a non
callable bond. And positive convexity is always good because we benefit in
either direction of interest rate movements.
So as the gamma, if delta is hedged, we should benefit from positive gamma
no matter price goes up or down.
Am I right or not?
If I'm right, why we want to hedge gamma? |
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a****5
发帖数: 2 | 22 Reference:
Here are the books I used for preparation. All the commons are just my
opinion.
1. "A Practical Guide to Quantitative Finance Interviews": You should be
fine with most of the standard
BT/Probability/Math/Option pricing questions if you truly UNDERSTAND every
part of the book.
2. "C++ Primer": It is too lengthy and is full of details. But it is very
comprehensive and you can find
everything there (before or after you are asked the questions :) )
3. "Effective C++": The first 10 ~ 20 it... 阅读全帖 |
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r**a
发帖数: 536 | 23 Why do you allow \tau=\infty? For american calls，the condition \tao\in [0,
T] is enough, isn't?
Further, if you change the payoff function \Phi(S(T)):= max{S(T)-K, 0}, you
change the whole thing in my opinion. Because here the function \Phi is a
convex function of S(T). But if you define \Phi(S(T)):= S(T)-K, \Phi is a
line function of S(T). A convex function is related to submartingale while
the line function is related to the martingale. They will affect the time
you reach the maximum of final ... 阅读全帖 |
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M*******r
发帖数: 165 | 24 how to value a derivative pay off 100R in five years, where R is the 1 year
interest rate observed in 4 years. What about pay off in 4 year? 6 year?
The answer is:
5 year: 100R_0*P(0,5)
4 year need convex adjustment: 100(R_0+c)P(0,4)
6 year: 100(R_0-c)P(0,4)
somebody explain the convex adjustment? Thx! |
|
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A**u
发帖数: 2458 | 26 面的Application developer职位，问的不少C++题目，都答上来了，但是2周了，还没
消息，基本确定被面试的老中给黑了
题目
1. Array Vs List
2. Set Vs unordered_set
3. vector.push_back()会发生啥
4. 为什么destructor不能扔出异常
5. new vs malloc
6. Pro-type of copy constructor
7. Pro-type of assignment operator
8. Bond的Duration
9. Bond的Convexity
除了Bond的convexity没答上，别的都很快给正确答案. 但是还给黑了
唉 叹息... |
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k*****n
发帖数: 117 | 27 Let V be the price to deliver $100 one year from now,
then V is a function of interest rate r.
The curve is convex.
V
|
|x
| x
| x
| x
| x
| x
| x
+-------------r
You want to compare E[V[R]] with V[5.0%]
Jensen's inequality will only give you E[V[R]] >= V[E[R]] = V[5.1%]
So you need to calculate actual price or calculate bond convexity at 5.0% to
compare.My guess is that E[V[R]] > V[5.0%], meaning current price is cheap
and you should take the money later. i.e. invest mo... 阅读全帖 |
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j******n
发帖数: 91 | 28 两个call. 一个constant volatility 40%, 一个stochastic volatility with mean
40%. 问两个哪个值钱。
option price is in most cases a convex function of volatility. 所以是由
convex property, constant volatility 的那个option 更值钱。对吗？ |
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S*********g
发帖数: 5298 | 29 This is strict.
Actually, it is a theorem in information theorem.
I is just the entropy of the {1,2,3,4,...,n}.
It is -E(log x) if you explain x_i as possibility of i.
Use the Jenson's inequality, E(f(x))>=f(Ex) if f is convex.
You will get that the entropy reaches the maximum when {i} is
uniformaly distributed, say x_i=1/n.
The maximun value is log(n)
Notice -Log is convex.
Then,
E log(1/[nx]) <= log(E(1/[nx]))<= log(1)=0
-E(log(x))<=E(log(n))=log(n). |
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a******y
发帖数: 2 | 30 Here is a rough provement.
Consider such a problem,
f(x) >= 0 and convex for x \in [a,b], a < 0 < b
f(x) = 0 otherwise.
\int_0^{-\infty} xf(x) dx = \int_0^{+\infty} xf(x)
S1 = \int_a^0 f(x) dx
S2 = \int_0^b f(x) dx
then S1 >= S2 * 4/5.
[PROVE]:
Let g(x) be the line passes (0, f(0)), (-2e, 0), where e > 0, and
S1 = \int_{-2e}^0 g(x) dx. Since f(x) convex, f(x) cross g(x) at most
once between -2e and 0, and
\int_0^{-\infty} xf(x) dx <= \int_0^{-2e} xg(x) dx.
It's not hard to show g(x) >= f(x) for |
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a******y
发帖数: 2 | 31 I do not have a very rigorous prove either. :)
Here are some explanations.
这个式子的几何意义我很清楚，可是如何从已知的条件里推出来呢？
- it seems it's not easy to prove 几何意义. :)
- f(x) is convex, that is f(tx + (1-t)y) >= tf(x) + (1-t)f(y)
for all x, y in (a,b), t in [0,1], It's not hard to prove
f(x) is continuous for x in (a,b).
- if |a| > |2e|, f(a) >= 0 > g(a), f(x) > g(x) by convex,
area under f(x) will be larger than area under g(x). impossible.
- if |a| = |2e|, f(a) >= 0 = g(a), it has to be f(x) = g(x), equali |
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c***z
发帖数: 6348 | 32 既然我们的优势是技术，劣势是资金，那么我们应该选择技术回报convex，资金回报
concave的行业（大部分公司正好相反，资金回报convex，技术和劳动回报concave）。
有这样的行业么？ |
|
|
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t**********t
发帖数: 12071 | 35 对了，你对discrete convex optimization可有研究？ |
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k*****r
发帖数: 21039 | 36 stephen boyd, "convex optimization"
书放在网上随便下。俺开课也用这套教材。 |
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e*u
发帖数: 10016 | 37 我认为我原帖里回复，提倡说既然社会发展方向就是由不同矢量方向的两个变量组成（
民）和（民的力量），普通百姓要想走在正确的方向，就要自强不息，跟制度没啥关系。
事实上按照我的公式，还可以列出来 （官*官的力量）运动方向，（商*商的力量）运
动方向，（中国人*中国人的力量）运动方向，（米犹*米犹的力量）运动方向。。。。
。。
这样子就能比较清楚理性的描述社会，进行建模。如果有翔实可靠的数据的话，还可以
根据PPP，GDP，CPI，用电量，发电机，铁路线路等等参数来描述各自力量的不同，由
此出现平衡点啦，attracive force,repulsive force,convex space啥啥啥的，倒是挺
好玩的。 |
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d*****g
发帖数: 4364 | 38 我日，你还敢跟我谈概率, 我就是靠这个吃饭的。
你废话了半天，意思就是根据你的假设 (对错很难说)，数学期望为负，所以去赌一下不划算吧。
但是你能根据中国这些年又进步，就得出共党是最优解吗？
你再回去复习一下. 把probability, random process, convex optimization 都看
看。 |
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w********l
发帖数: 11447 | 39 我日，你这回答就好比，我问你吃过了吗？ 你呀特激动的回答了一句，我刚从厕所回
来，你说呢？
你扯那么多random process和convex optimization，跟我这有个毛关系啊。我跟你说
posterior probability,你跟我整random, 我晕死。你们老板怎么找你这么个糊弄人的
所谓专家？ 还不如给我这样业余的人做。
俺这么直接的比喻，你都看不懂，难怪唧唧歪歪，如果没有土共，会如何如何。你好好
思考一下，到底愿不愿意抛硬币，让自己要么帅点，要么直接做狗？思考晚了，再来谈
你这个假如没有土共的问题。
下不划算吧。 |
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k********a
发帖数: 446 | 40 传统机器学习/人工智能（SVM，kernel，convex optimization，topic model，etc）
搞了很多年，对语音识别这个最基本的问题基本没有什么大贡献，从这个角度来说近期
的DL语音识别的工作使得识别率前进了一大步，推进到了工业级的应用，进步还是很实
在的。至于记者们瞎引深，那也没办法。最近 IEEE spectrum interviewed both
Michael Jordan 和 Yann LeCun, 很有意思。 |
|
发帖数: 1 | 41 当时要选优化课的，我去economic系专门选了几门 优化课，其中就是Convex
Optimization
现在是Steven Boyd出书了，但当时我上经济学的时候，不是他在教课，一直在EE，我
选过好几门EE是他教的，他偏重于数学的，我这方面有优势，所以他的体系和出题思路
我很熟悉，就一路选了他几乎所有的大课 |
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t******l
发帖数: 10908 | 42 或者马工理论的说，决策理论里面的不管是 graph-based dynamic programming 比如
dijkstra's algorithm，还是 linear programming high-dimension convex polytope
based algorithm，或者 nonlinear conjugated gradient method 等等等等。。。最
低的底线是你小黄人的小命 cost 不能为零不是？
你小黄人的小命 cost 为零，additive identity，加了跟不加一样，啥算法都是个球
不是？
。。 |
|
b***y
发帖数: 14281 | 43 Convexity is just one special kind of non-linearity. |
|
h*******i
发帖数: 4386 | 44 non-convexity is the main challenge.
不过这一套还是毛子厉害，过去50年，也就是毛子的barrier method算是个突破，别的
都是然并暖。 |
|
b***y
发帖数: 14281 | 45 Non-concexity meaning instability?
: non-convexity is the main challenge.
: 不过这一套还是毛子厉害，过去50年，也就是毛子的barrier method算是个突破
，别的
: 都是然并暖。
|
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d********m
发帖数: 3662 | 46 optimization rules
statistics is pretty much a problem of convex optimization |
|
s***h
发帖数: 487 | 47 Hessian matrix positive definite 说明是 convex function，只有一个极致。
但 Nonconvex function 也可以有极致，还可以很多。
: 笔误了，是Hessian matrix。
: 不过你没说出我问题的答案，应该是找Hessian matrix's determinant, which
is a
: matrix here, is positive semidefinite or not. If it is then there is
an
: extremity.
: Matrix
|
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s***h
发帖数: 487 | 48 Nonconvex function 可以有极致，也可以没有，俗话说：inconclusive 。。。 如果
没有记错
: Hessian matrix positive definite 说明是 convex function，只有一个极致。
: 但 Nonconvex function 也可以有极致，还可以很多。
: is a
: an
|
|
f**********d
发帖数: 4960 | 49 你弄错了，梯度为0，hessian不正定是鞍点，说明此驻点不是极值点。
hessian矩阵是对应一维函数时在驻点的二阶导数的*绝对值*，所以hessian正定包括
convex和concave两种情况。
which |
|
s***h
发帖数: 487 | 50 另外的分歧，你说的可能是对单个点而言，我说的是对一个区间或者整个函数域而言。
: 你弄错了，梯度为0，hessian不正定是鞍点，说明此驻点不是极值点。
: hessian矩阵是对应一维函数时在驻点的二阶导数的*绝对值*，所以hessian正定
包括
: convex和concave两种情况。
: which
|
|
