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
打开 →题目1996 · 数学
Convex Execution Cost With a Capacity Barrier 1
A schedule pays a quadratic cost but also faces a blow-up term as it nears a hard capacity cap. Show that f(q) = 1 q^2 + 2/(1-q) is strictly convex on q<1.
打开 →题目2020 · 数学
Convex Margin Penalty With a Soft Capacity Wall 25
The margin term grows smoothly but sharply as the leverage coordinate approaches its cap. Show that r(x) = -ln(1-1x) + 3x^2 is convex on x < 1.
打开 →题目2039 · 数学
Convex Penalty for Mixed Schedules 19
Two execution schedules have penalties phi(q_1) and phi(q_2) under a convex phi. What does Jensen say about a random 50-50 mix versus the penalty at the average size?
打开 →题目1998 · 数学
Convex Portfolio Penalty With Aggregate Exposure 3
Each sleeve has its own quadratic penalty, and the whole book also pays for aggregate balance-sheet usage. Prove that F(w_1,w_2,w_3) = 2w_1^2 + 3w_2^2 + 5w_3^2 + 1(w_1+w_2+w_3)^2 is convex.
打开 →题目2012 · 数学
Convexity of a Tighter Utilization Penalty 17
The utilization cap is tighter, but the same barrier argument applies. Show that r(x) = -ln(1-3x) + 2x^2 is convex on x < 0.333333.
打开 →题目2005 · 数学
Convexity of a Two-Asset Quadratic With a Balance-Sheet Term 10
Show that H(w_1,w_2)=2w_1^2+5w_2^2+3(w_1+w_2)^2 is convex.
打开 →题目2019 · 数学
Convexity of Logistic Loss 24
Show that ell(z)=ln(1+e^{-z}) is convex on R.
打开 →题目2424 · 机器学习
Convexity of the Log-Cosh Loss 4
Show that ell(r)=ln cosh(r) is convex in the residual r.
打开 →题目4448 · 机器学习
Correlation Shock Benefit 8
An equal-weight composite combines two standardized signals. If their correlation drops from 0.6 to 0.2, by how much does the composite standard deviation fall?
打开 →题目2640 · 机器学习
Cosine Decay Schedule 12
A learning rate decays from eta_max to eta_min over T steps using cosine annealing. What is eta_t at step t?
打开 →题目1600 · 数学
Coupling Needed for Equal Optimizer Coordinates 5
For f(x) = 0.5 x^T A(t) x - b^T x with A(t) = [[5,t],[t,4]] and b = (6,2), for what t does the optimizer satisfy x1 = x2?
打开 →题目2427 · 机器学习
Decision Threshold Under Asymmetric Classification Cost
A false negative costs 5 and a false positive costs 1. If p is the predicted probability of the positive class, above what threshold should you classify as positive?
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