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 →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 →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 →题目5905 · 概率
Kelly with a Cash-Reserve Floor
You may stake on an even-money coin with win probability $p=0.8$, but a liquidity rule requires you to keep at least half your total wealth in untouched cash at all times, so the staked fraction satisfies $f\le 0.5$. Set up the constrained maximization of expected log-growth, use
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