题目3363 · 数学
When KKT Is Not Enough by Itself
Give one reason why solving the KKT equations in a nonconvex problem may fail to identify the global optimum.
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中文题目题目3361 · 数学
Why Convexity Makes KKT So Powerful
Why do KKT conditions become sufficient, not just necessary, in many convex optimization problems?
打开 →课程凸优化 · 最优化
拉格朗日对偶与 KKT 条件
开篇场景（Hook）：PM 真正想要的那个数 上海一家中型私募的 PM 周一早盘正在跟风控拉锯：当前组合的总杠杆（gross leverage）顶在 200% 的合规上限，他想申请抬到 220%。风控的问题不是「能不能」，而是「值不值」——多 20 个百分点能换多少边际信息比率（marginal information ratio）？答案其实早就躺在凸求解器...
打开 →题目3346 · 数学
Closest Point to the Origin Above a Budget Line
Minimize $x^2+y^2$ subject to $x+y\ge 1$. Find $(x^*,y^*)$ and the optimal KKT multiplier for the inequality $g(x,y)=1-x-y\le 0$.
打开 →题目3356 · 数学
Feasible Center Means Zero Multiplier
Consider minimizing $(x-1)^2+(y-1)^2$ subject to $x+y\ge 1$. At the optimum, is the inequality active or inactive, and what does KKT imply about $\lambda$?
打开 →题目3360 · 数学
High y Coordinate Makes the Constraint Slack
Consider minimizing $(x--1)^2+(y-3)^2$ subject to $x+y\ge 1$. At the optimum, is the inequality active or inactive, and what does KKT imply about $\lambda$?
打开 →题目3345 · 数学
Negative Target, Mildly Active Floor
Minimize $(x--2)^2$ subject to $x\ge -1$. Using the KKT form $g(x)=-1-x\le 0$, find the optimizer $x^*$ and the optimal multiplier $\lambda$.
打开 →题目3358 · 数学
Offset Point Already Feasible
Consider minimizing $(x-2)^2+(y--1)^2$ subject to $x+y\ge 0$. At the optimum, is the inequality active or inactive, and what does KKT imply about $\lambda$?
打开 →题目3341 · 数学
Projected Optimum Above the Floor
Minimize $(x-2)^2$ subject to $x\ge 0$. Using the KKT form $g(x)=0-x\le 0$, find the optimizer $x^*$ and the optimal multiplier $\lambda$.
打开 →题目3351 · 数学
Weighted Quadratic with Cheap x and Expensive y
Minimize $1x^2+3y^2$ subject to $x+y\ge 4$. Find $(x^*,y^*)$ and the KKT multiplier.
打开 →题目3365 · 数学
Why Slater-Type Regularity Matters
Why do regularity conditions such as Slater's condition matter when applying KKT?
打开 →模块2.5.1 · 数学与统计能力 · 最优化
凸优化
凸优化 · KKT · 梯度下降 · 组合约束优化
打开 →课程实践约束与压力测试 · 组合构建与风险
仅做多与 MV QP 中的线性约束
某沪深300指增公募的高级量化研究员,把 4.4.1 的均值方差闭式解 w = (1/gamma) Sigma^ (mu lambda 1) 直接套到她管理的 30 只 CSI 300 成分股核心仓上。闭式解给出的结果:招商银行 600036 做空 300%、宁德时代 300750 多头 +250%、组合 78% 的仓位扎堆在前三只动量名上。她的产品合同写得...
打开 →课程凸优化 · 最优化
凸优化问题与标准形式
开篇场景（Hook）：一位 PM 的两份委托书 周一上午，你在一家 沪深300 指数增强 私募 基金的研究台收到两份新增的客户委托书。第一份要求满仓多头、公式、公式、行业偏离度上限 ±3%（一组线性不等式）——干净的二次规划（quadratic program, QP）：二次目标 + 仿射约束，求解器十秒出结果。第二份加了一句「持仓数不得超过 50 只」，可...
打开 →课程凸优化 · 最优化
组合优化与锥规划建模
周五下午三点，你在某 公募 基金管理一只 沪深300 指数增强（CSI 300 enhanced index）产品。当前基金合同把年化 跟踪误差（tracking error）上限设在 300 bp。求解器把当日再平衡的解返回过来——主仓位都合理，但对偶价格表里 跟踪误差 约束的乘子写着 公式 bp。翻译成 PM 听得懂的语言：若把上限从 300 bp 放到...
打开 →题目3362 · 数学
Complementary Slackness as an Economic Statement
Explain complementary slackness in plain language to a PM who thinks of constraints as scarce resources.
打开 →题目1793 · 统计
Elastic Net Grouping
Two features are almost duplicates but both are economically meaningful. Why does Elastic Net often behave better than pure Lasso here?
打开 →题目2501 · 机器学习
Equivalent L2 Radius From a Ridge Solution 6
If the ridge optimum in R^p is beta_hat_lambda, what radius t makes it also solve the constrained problem min RSS(beta) subject to ||beta||_2 <= t?
打开 →题目2500 · 机器学习
Equivalent Lambda for a Target Ridge Shrinkage Ratio 5
In an orthogonal coordinate, ridge shrinks beta_ols by the factor d/(d+lambda). What lambda yields a shrinkage ratio r in (0,1)?
打开 →题目2512 · 机器学习
Lasso Activation Threshold Numerically 18
In an orthogonal coordinate with z = 7, what is the smallest lambda that forces the lasso coefficient to zero?
打开 →题目1776 · 统计
Lasso Threshold Calibration 1
A standardized lasso fit has score vector (4.1, 2.3, 1.7). What is the smallest lambda that makes every coefficient exactly zero?
打开 →题目1779 · 统计
Lasso Threshold Calibration 4
In an orthonormal lasso update, a coordinate has score z = -3.2 and penalty lambda = 0.7. What coefficient results after soft-thresholding?
打开 →题目2503 · 机器学习
MAP Interpretation of Ridge 10
Under a Gaussian-noise linear model, what Gaussian prior on beta makes ridge the MAP estimator?
打开 →题目2505 · 机器学习
Ridge Never Flips Sign in the Orthogonal One-Feature Case 12
In the orthogonal one-feature case with z = x^T y, why does ridge preserve the sign of z for every lambda >= 0?
打开 →题目2509 · 机器学习
Ridge Norm Shrinks Monotonically With Lambda 15
Why should the ridge solution norm typically decrease as lambda increases?
打开 →题目3364 · 数学
Shadow Price Interpretation
Why is the optimal multiplier often interpreted as the marginal value of relaxing or tightening a constraint?
打开 →题目2515 · 机器学习
Why Small Lambda Means Weak Regularization 20
Why does a very small lambda leave the regularized solution close to OLS?
打开 →题目1795 · 统计
Why the One-SE Rule Is Conservative
Why do practitioners often prefer the one-standard-error rule over the absolute CV minimizer when selecting a regularization parameter?
打开 →题目2510 · 机器学习
Zero Lambda Recovers OLS 16
Why do ridge and lasso both reduce to OLS when their regularization parameter is set to zero?
打开 →课程凸优化 · 最优化
凸集与凸函数
开篇场景（Hook）：基数约束如何把 MVO 从 QP 推到 MIP 周二下午，你在一家百亿规模的私募（private fund）量化部门里，给沪深300 成分股做一只全仓做多组合。脚本是教科书版本的均值方差优化（mean variance optimization, MVO）：最小化 公式，约束 公式、公式。CVXPY 在 80ms 内返回全局最优。你顺手...
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