docs: added simplified optimization problem formulation doc

.design_docs/optimizer-mip-formulation.md

@@ -9,7 +9,7 @@ For example - `avg_over_time(data[5m]) / quantile_over_time(0.5, data[5m])` is a
 
 ## Assumptions
 
-**No cross-RQE result reuse.** $f_a = \sum_{r \in R_a} 1/T_r$ counts every RQE invocation of AQE $a$ as independent query work. In practice, if two RQEs referencing the same AQE co-fire (execute at the same moment), the system could compute $a$ once and share the result. This is not modeled. To do so would require: phase/offset information per RQE (to determine co-firing frequency), and a sharing decision variable (since co-firing RQEs may still query different time windows and cannot always share). The current model overestimates query cost, which is conservative.
+**Cross-RQE reuse is not modeled in query cost.** $f_a = \sum_{r \in R_a} 1/T_r$ counts every RQE invocation of AQE $a$ as independent query work. In practice, if two RQEs referencing the same AQE co-fire (execute at the same moment), the system could compute $a$ once and share the result. This is not modeled. To do so would require: phase/offset information per RQE (to determine co-firing frequency), and a sharing decision variable (since co-firing RQEs may still query different time windows and cannot always share). The current model overestimates query cost, which is conservative.
 
 ## Inputs
 

.design_docs/optimizer-simplified-formulation.md

@@ -0,0 +1,63 @@
+# Simplified formulation of the optimization formulation
+
+## Mental model
+
+Given a workload of repeating queries over time-series data, we want to map the query workload to an ensemble of summarization strategies to deploy.
+Each query performs a certain aggregation over a certain time range over a given metric.
+Each summarization strategy takes an input stream of metrics and computes summaries (exact or approximate) on it in a streaming manner. When queries are received, they are mapped to specific summaries, and answered using them (either directly from a single summary or by manipulating summaries such as merging them)
+
+![Mental model diagram](mental-model-diagram.png)
+
+
+## Simplifying assumptions
+
+- Data arrives at a regular `data_ingestion_interval`. E.g. for Prometheus, this will be the Prometheus scrape interval
+- Each query repeats at the same time interval $T_r$.
+- $T_r$ is a multiple of `data_ingestion_interval`
+
+## Inputs
+
+| Symbol | Source | Definition |
+|--------|--------|-----------|
+| $R = \{r_1, \ldots, r_n\}$ | User-specified | Set of RQEs (Repeating Query Expressions) |
+| $T_r$ | User-specified | Time interval between repetition of each Query Expression |
+| $\varepsilon_r$ | User-specified | Accuracy tolerance for RQE $r$ |
+| $\text{SLA}_r$ | User-specified | Latency requirement for RQE $r$ |
+| $S = \{s_1, \ldots s_n\}$ | Expert-defined | Set of summarization strategies |
+| `ingest_cost(s)` | Expert-defined | Cost of ingesting data into a summary $s$ |
+| `query_cost(s, r)` | Expert-defined | Cost of answering RQE $r$ from summary $s$ |
+
+## Internal Variables
+
+| Symbol | Derived from | Definition |
+|--------|-------------|-----------|
+| $f_r$ | Query workload | $1/T_r$ — query rate for RQE $r$ (queries/sec) |
+
+## Outputs
+
+| Symbol | Definition |
+|--------|-----------|
+| $y_s \in \{0,1\}$ | 1 if summarization strategy $s$ is deployed. |
+| $x_{r,s} \in \{0,1\}$ | 1 if RQE $r$ is served by strategy $s$. |
+
+## Objective
+
+$$\min \sum_{s \in S} y_s \cdot \mathrm{ingest\_cost}(s) \ + \ \sum_{r \in R,\ s \in S} x_{r,s} \cdot f_r \cdot \mathrm{query\_cost}(s, r)$$
+
+Both terms are cost rates (cost/sec). $f_r$ converts the per-query `query_cost` into a rate commensurate with the continuously-accruing `ingest_cost`.
+
+## Constraints
+
+$$\sum_{s \in S} x_{r,s} = 1 \qquad \forall r \in R \tag{1}$$
+
+$$x_{r,s} \leq y_s \qquad \forall r \in R,\ s \in S \tag{2}$$
+
+$$x_{r,s},\ y_s \in \{0,1\} \tag{3}$$
+
+**(1)** Every RQE is served by exactly one strategy. **(2)** An RQE cannot be served by a strategy that is not deployed. **(3)** Integrality.
+
+## Challenges
+
+1. How to define the set of available summarization strategies $S$?
+2. How to infer if a particular RQE $r$ can be feasibly answered using a summarization strategy $s$?
+3. How to efficiently infer `ingest_cost` and `query_cost`, for different summarization strategies, queries, and data shapes/distributions?