Introduction to mathematical programming
4th Edition
ISBN: 9780534359645
Author: Jeffrey B. Goldberg
Publisher: Cengage Learning
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Chapter 4.7, Problem 6P
Explanation of Solution
Basic solutions not being optimal
- Any other basic solution will have s1 and/or s2 as a basic variable.
- Since no right hand side is 0, if s1 or s2 is pivoted in it will be positive...
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Here are two diagrams. Make them very explicit, similar to Example Diagram 3 (the Architecture of MSCTNN).
graph LR subgraph Teacher_Model_B [Teacher Model (Pretrained)] Input_Teacher_B[Input C (Complete Data)] --> Teacher_Encoder_B[Transformer Encoder T] Teacher_Encoder_B --> Teacher_Prediction_B[Teacher Prediction y_T] Teacher_Encoder_B --> Teacher_Features_B[Internal Features F_T] end subgraph Student_B_Model [Student Model B (Handles Missing Labels)] Input_Student_B[Input C (Complete Data)] --> Student_B_Encoder[Transformer Encoder E_B] Student_B_Encoder --> Student_B_Prediction[Student B Prediction y_B] end subgraph Knowledge_Distillation_B [Knowledge Distillation (Student B)] Teacher_Prediction_B -- Logits Distillation Loss (L_logits_B) --> Total_Loss_B Teacher_Features_B -- Feature Alignment Loss (L_feature_B) --> Total_Loss_B Partial_Labels_B[Partial Labels y_p] -- Prediction Loss (L_pred_B) --> Total_Loss_B Total_Loss_B -- Backpropagation -->…
Please provide me with the output image of both of them . below are the diagrams code
I have two diagram :
first diagram code
graph LR subgraph Teacher Model (Pretrained) Input_Teacher[Input C (Complete Data)] --> Teacher_Encoder[Transformer Encoder T] Teacher_Encoder --> Teacher_Prediction[Teacher Prediction y_T] Teacher_Encoder --> Teacher_Features[Internal Features F_T] end subgraph Student_A_Model[Student Model A (Handles Missing Values)] Input_Student_A[Input M (Data with Missing Values)] --> Student_A_Encoder[Transformer Encoder E_A] Student_A_Encoder --> Student_A_Prediction[Student A Prediction y_A] Student_A_Encoder --> Student_A_Features[Student A Features F_A] end subgraph Knowledge_Distillation_A [Knowledge Distillation (Student A)] Teacher_Prediction -- Logits Distillation Loss (L_logits_A) --> Total_Loss_A Teacher_Features -- Feature Alignment Loss (L_feature_A) --> Total_Loss_A Ground_Truth_A[Ground Truth y_gt] -- Prediction Loss (L_pred_A)…
Chapter 4 Solutions
Introduction to mathematical programming
Ch. 4.1 - Prob. 1PCh. 4.1 - Prob. 2PCh. 4.1 - Prob. 3PCh. 4.4 - Prob. 1PCh. 4.4 - Prob. 2PCh. 4.4 - Prob. 3PCh. 4.4 - Prob. 4PCh. 4.4 - Prob. 5PCh. 4.4 - Prob. 6PCh. 4.4 - Prob. 7P
Ch. 4.5 - Prob. 1PCh. 4.5 - Prob. 2PCh. 4.5 - Prob. 3PCh. 4.5 - Prob. 4PCh. 4.5 - Prob. 5PCh. 4.5 - Prob. 6PCh. 4.5 - Prob. 7PCh. 4.6 - Prob. 1PCh. 4.6 - Prob. 2PCh. 4.6 - Prob. 3PCh. 4.6 - Prob. 4PCh. 4.7 - Prob. 1PCh. 4.7 - Prob. 2PCh. 4.7 - Prob. 3PCh. 4.7 - Prob. 4PCh. 4.7 - Prob. 5PCh. 4.7 - Prob. 6PCh. 4.7 - Prob. 7PCh. 4.7 - Prob. 8PCh. 4.7 - Prob. 9PCh. 4.8 - Prob. 1PCh. 4.8 - Prob. 2PCh. 4.8 - Prob. 3PCh. 4.8 - Prob. 4PCh. 4.8 - Prob. 5PCh. 4.8 - Prob. 6PCh. 4.10 - Prob. 1PCh. 4.10 - Prob. 2PCh. 4.10 - Prob. 3PCh. 4.10 - Prob. 4PCh. 4.10 - Prob. 5PCh. 4.11 - Prob. 1PCh. 4.11 - Prob. 2PCh. 4.11 - Prob. 3PCh. 4.11 - Prob. 4PCh. 4.11 - Prob. 5PCh. 4.11 - Prob. 6PCh. 4.12 - Prob. 1PCh. 4.12 - Prob. 2PCh. 4.12 - Prob. 3PCh. 4.12 - Prob. 4PCh. 4.12 - Prob. 5PCh. 4.12 - Prob. 6PCh. 4.13 - Prob. 2PCh. 4.14 - Prob. 1PCh. 4.14 - Prob. 2PCh. 4.14 - Prob. 3PCh. 4.14 - Prob. 4PCh. 4.14 - Prob. 5PCh. 4.14 - Prob. 6PCh. 4.14 - Prob. 7PCh. 4.16 - Prob. 1PCh. 4.16 - Prob. 2PCh. 4.16 - Prob. 3PCh. 4.16 - Prob. 5PCh. 4.16 - Prob. 7PCh. 4.16 - Prob. 8PCh. 4.16 - Prob. 9PCh. 4.16 - Prob. 10PCh. 4.16 - Prob. 11PCh. 4.16 - Prob. 12PCh. 4.16 - Prob. 13PCh. 4.16 - Prob. 14PCh. 4.17 - Prob. 1PCh. 4.17 - Prob. 2PCh. 4.17 - Prob. 3PCh. 4.17 - Prob. 4PCh. 4.17 - Prob. 5PCh. 4.17 - Prob. 7PCh. 4.17 - Prob. 8PCh. 4 - Prob. 1RPCh. 4 - Prob. 2RPCh. 4 - Prob. 3RPCh. 4 - Prob. 4RPCh. 4 - Prob. 5RPCh. 4 - Prob. 6RPCh. 4 - Prob. 7RPCh. 4 - Prob. 8RPCh. 4 - Prob. 9RPCh. 4 - Prob. 10RPCh. 4 - Prob. 12RPCh. 4 - Prob. 13RPCh. 4 - Prob. 14RPCh. 4 - Prob. 16RPCh. 4 - Prob. 17RPCh. 4 - Prob. 18RPCh. 4 - Prob. 19RPCh. 4 - Prob. 20RPCh. 4 - Prob. 21RPCh. 4 - Prob. 22RPCh. 4 - Prob. 23RPCh. 4 - Prob. 24RPCh. 4 - Prob. 26RPCh. 4 - Prob. 27RPCh. 4 - Prob. 28RP
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- Here is a clear background and explanation of the full method, including what each part is doing and why. Background & Motivation Missing values: Some input features (sensor channels) are missing for some samples due to sensor failure or corruption. Missing labels: Not all samples have a ground-truth RUL value. For example, data collected during normal operation is often unlabeled. Most traditional deep learning models require complete data and full labels. But in our case, both are incomplete. If we try to train a model directly, it will either fail to learn properly or discard valuable data. What We Are Doing: Overview We solve this using a Teacher–Student knowledge distillation framework: We train a Teacher model on a clean and complete dataset where both inputs and labels are available. We then use that Teacher to teach two separate Student models: Student A learns from incomplete input (some sensor values missing). Student B learns from incomplete labels (RUL labels missing…arrow_forwardhere is a diagram code : graph LR subgraph Inputs [Inputs] A[Input C (Complete Data)] --> TeacherModel B[Input M (Missing Data)] --> StudentA A --> StudentB end subgraph TeacherModel [Teacher Model (Pretrained)] C[Transformer Encoder T] --> D{Teacher Prediction y_t} C --> E[Internal Features f_t] end subgraph StudentA [Student Model A (Trainable - Handles Missing Input)] F[Transformer Encoder S_A] --> G{Student A Prediction y_s^A} B --> F end subgraph StudentB [Student Model B (Trainable - Handles Missing Labels)] H[Transformer Encoder S_B] --> I{Student B Prediction y_s^B} A --> H end subgraph GroundTruth [Ground Truth RUL (Partial Labels)] J[RUL Labels] end subgraph KnowledgeDistillationA [Knowledge Distillation Block for Student A] K[Prediction Distillation Loss (y_s^A vs y_t)] L[Feature Alignment Loss (f_s^A vs f_t)] D -- Prediction Guidance --> K E -- Feature Guidance --> L G --> K F --> L J -- Supervised Guidance (if available) --> G K…arrow_forwarddetails explanation and background We solve this using a Teacher–Student knowledge distillation framework: We train a Teacher model on a clean and complete dataset where both inputs and labels are available. We then use that Teacher to teach two separate Student models: Student A learns from incomplete input (some sensor values missing). Student B learns from incomplete labels (RUL labels missing for some samples). We use knowledge distillation to guide both students, even when labels are missing. Why We Use Two Students Student A handles Missing Input Features: It receives input with some features masked out. Since it cannot see the full input, we help it by transferring internal features (feature distillation) and predictions from the teacher. Student B handles Missing RUL Labels: It receives full input but does not always have a ground-truth RUL label. We guide it using the predictions of the teacher model (prediction distillation). Using two students allows each to specialize in…arrow_forward
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