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?
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中文题目课程面向最优化的微积分 · 线性代数与微积分
一阶最优性与凸性
Hook：两个看起来都「会优化」的求解器 上海某私募基金的两位研究员同时打开 Python，一位在跑一个标的为沪深300 成分股、目标为均值方差优化（mean variance optimization）的组合优化（portfolio optimization）问题，另一位在调一个三层的因子神经网络。两人用的迭代算法是同一份梯度下降代码，第一位 200 步就...
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...
打开 →模块4.4.3 · 量化全流程 · 组合构建与风险
实践约束与压力测试
portfolio-optimization · constrained-mv · quadratic-programming · qp · cvxpy · osqp · long-only · leverage-cap
打开 →模块2.5.2 · 数学与统计能力 · 最优化
迭代法与正则化方法
optimization · gradient-descent · line-search · convergence · iterative-methods · newton-method · quasi-newton · bfgs
打开 →模块2.4.2 · 数学与统计能力 · 线性代数与微积分
面向最优化的微积分
calculus · gradient · directional-derivative · optimization · chain-rule · jacobian · backpropagation · taylor-expansion
打开 →课程实践约束与压力测试 · 组合构建与风险
交易成本感知优化与换手率预算
某沪深300指增私募的中级量化研究员,用 L1 的「无成本」约束 MV 优化器跑 30 只 CSI 300 行业龙头基础上的 12 1 截面动量信号,样本内纸面 Sharpe(paper Sharpe)= 1.4。她把同样的换仓单丢进自家交易台的事后成本归因系统,扣掉佣金、印花税、半价差(half spread)和 Almgren Chriss 市场冲击之后...
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
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 →课程实践约束与压力测试 · 组合构建与风险
端到端组合构建管线、压力测试与部署
某沪深300指增私募的策略部署组组长周一早会带着三份交付物走进风控委员会。PM 刚审批通过一只新主动股票策略,研究组把 4.2 alpha 管线(截面动量 + 质量 + 价值的复合 alpha,样本内 IR 约 0.5)、4.3 因子暴露矩阵 B (Barra 风格 5 个 + 中信一级行业 10 个 + 国家因子)、4.4.2 Barra 风险模型 (Si...
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 →课程实践约束与压力测试 · 组合构建与风险
鲁棒优化与分布鲁棒投资组合优化
某沪深300指增私募的中级量化研究员把 L2 的成本感知优化器跑了三年。样本内纸面 Sharpe = 1.4,实盘 Sharpe = 0.7。她把回测净值拆开,发现两件事:12 1 截面动量 mu hat 的标准误差是均值的 3 5 倍——Chopra Ziemba(1993)《The effect of errors in means, variances...
打开 →题目2031 · 数学
Backing Out the Stress State From a Reciprocal Plug-In Score 11
A funding-buffer score uses phi(L)=1/(1+L). Suppose leverage L equals 0 with probability 1/2 and H with probability 1/2. If phi(E[L]) = 1/3, what is H and what is E[phi(L)]?
打开 →题目1962 · 数学
Balanced Exponential Penalties 17
A desk minimizes J(x)=6 e^x + 3 e^{-2x}. What x is optimal?
打开 →题目1966 · 数学
Balanced Softplus Tradeoff 21
The linear reward and saturation penalty balance exactly at a central point. The desk maximizes K(x) = 2 x - 4 ln(1+e^x). What x is optimal?
打开 →题目2008 · 数学
Barrier Cost With Strong Quadratic Curvature 13
Both the quadratic inventory term and the capacity wall are active sources of curvature. Show that f(q) = 4 q^2 + 3/(1-q) is strictly convex on q<1.
打开 →题目2032 · 数学
Barrier Score Gap for a Two-State Utilization Model 12
Let u(x)=-ln(1-x) on x<1. Suppose U equals 0 with probability 1/2 and 3/4 with probability 1/2. Compute E[u(U)] and u(E[U]).
打开 →题目2634 · 机器学习
Batch-Average Gradient 9
If the minibatch loss is the average L = (1/B) sum_{i=1}^B L_i, derive dL/dw in terms of the per-example gradients.
打开 →题目5899 · 概率
Betting Kelly on the Wrong Probability
An even-money coin truly wins with probability $p=0.55$, but you overestimate it as $\hat p=0.65$ and bet the Kelly fraction implied by your estimate. What is your actual long-run expected log-growth rate per round? Compare it to the growth you would have earned betting the corre
打开 →题目1951 · 数学
Bid-Ask Skew Balance 6
On one quote axis, the maker gets more value from aggressive bids than from aggressive offers. A market maker chooses a skew x in (-1,1) to maximize G(x) = 5 ln(1+x) + 3 ln(1-x). What skew is optimal?
打开 →题目2675 · 机器学习
Break-Even Hit Rate After Trading Costs
A directional model earns +1 unit on a correct trade and -1 unit on an incorrect trade before costs. Each round trip also pays a cost of 0.08 units regardless of outcome. What hit rate $p$ makes expected net PnL zero?
打开 →题目5955 · 概率
Buy Random or Buy the Missing One
You need all 5 types and currently hold 4 distinct types (exactly one type missing). Each round you may either (a) buy a random coupon for $1 (uniform over all 5 types), or (b) directly buy your missing type from a reseller for $5. Acting optimally to minimize expected total futu
打开 →题目5903 · 概率
Capping the Single-Bet Drawdown
You bet a fraction $f$ of wealth on an even-money coin with win probability $p=0.65$, but a risk rule forbids any single losing bet from cutting your wealth by more than $20\%$. What fraction should you bet, and for which win probabilities $p$ does this drawdown rule actually con
打开 →题目1956 · 数学
Carry Drag Versus Quadratic Inventory 11
A carry term explodes as the state approaches -1, so the desk cannot simply push x downward. Minimize H(x) = 4 x^2 + 9/(1+x) over x > -1.
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2045 · 数学
Comparing Two Equal-Mean Schedules Under an Alternative Surcharge 25
A utilization surcharge is c(q)=1/(2-q) on q<2. Schedule A is deterministic with Q=1. Schedule B uses Q=1/2 or 3/2 with probability 1/2 each. Compute E[c(Q)] for Schedule B and c(E[Q]) for the shared mean.
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2034 · 数学
Conditional Jensen Lower Bound 14
If phi is convex, what inequality holds between E[phi(X)|F] and phi(E[X|F]) almost surely?
打开 →题目5898 · 概率
Continuous Kelly for Normal Returns
Each round you allocate a fraction $f$ of wealth to a position whose one-period return $R$ is approximately normal with small mean $\mu>0$ and variance $\sigma^2$ (with $\mu^2\ll\sigma^2$), so post-round wealth is multiplied by $1+fR$. Using a second-order expansion of the log, d
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