combinatorial-optimization-using-comparison-oracles/theorem-general-linear-optimization.md

# Research Challenge: Linear optimization via comparisons and conic dimension

> **Source paper:** Combinatorial Optimization using Comparison Oracles  
> **Reference:** [arXiv:2511.15142](https://arxiv.org/abs/2511.15142) — Theorem general.linear.optimization  
> **Task:** Prove the theorem stated below. This document is self-contained: all definitions and notation needed to understand the statement are included. The proof is deliberately omitted.

## Overview

For any point set with conic dimension k and unknown linear weights, one can find the minimizer of the hidden linear objective using only O(k log k log|P|) comparison queries between candidate points.

## Definitions and setup

The following definitions and notation (quoted from the source paper) are referenced by the theorem.

### conic dimension (\CDfull)  *(— Definition, Section 'Envelopes and the CDfull of Point Sets')*

> The \CDfull of a set $\cY \subseteq \RR^d$, denoted $\CDmath(\cY)$, is the largest integer $k$ for which there exists a length-$k$ conically independent sequence $\sigma = (y_1, \dots, y_k) \in \cY^k$.

### comparison oracle  *(— Abstract)*

> we consider the weaker and more robust comparison oracle, which for any two feasible sets $S, T \in F$, reveals only if $w(S)$ is less than/equal to/greater than $w(T)$.

### \ip{\cdot,\cdot} (inner product)  *(— Abstract)*

> our goal is to find $S^* = \arg\min_{S \in F} \langle w,\mathbbm{1}_S \rangle$ ... we are given a value oracle which allows us to evaluate $w(S) := \langle w, \mathbbm{1}_S \rangle$ for any $S \in F$

### \cone(V) (cone generated by V)  *(— Section 'The Certification Problem')*

> given a set $V \sse \RR^d$ of vectors, the cone generated by $V$ is the set of all vectors that can be written as positive linear combinations of vectors from $V$; this is denoted by $\cone(V)$. Formally, $ \cone(V) = \left\{x \in R^d : x = \sum_{v \in V} \alpha_v \, v, \; \alpha_v \geq 0\right\}. $

### conically independent sequence  *(— Section 'Envelopes and the CDfull of Point Sets', Definition)*

> The points are conically independent if, for each $t \in \{2, \ldots, k\}$, \[ y_t - y_1 \notin \envelope(\sigma^{1:t-1}), \] where $\sigma^{1:t}$ denotes the prefix consisting of the first $t$ elements of $\sigma$.

### y-certificate  *(— Section 'The Certification Problem')*

> A $y$-certificate is a collection $\cC$ of index pairs $(i,j)$ such that for all $w \in R^d$, we have that $\left(\langle w, x_i - x_j\rangle \geq 0 \quad \forall (i,j)\in \cC\right) \Rightarrow \left(\langle w, x - y\rangle \geq 0 \quad \forall x \in \cP\right)$

### valid certificate  *(— Section 'The Certification Problem')*

> A certificate $\cC$ is valid if $\langle w^*, x_i - x_j \rangle \geq 0$ for all $(i,j) \in \cC$.


## The challenge

**Hypotheses / assumptions:**

- $\cP \subseteq \RR^d$ with conic dimension $k$
- unknown $w^* \in \RR^d$

**Statement to prove (Theorem general.linear.optimization):**

> There is an algorithm that, for any point set $\cP \sse \RR^d$ with \CDfull $k$ and unknown weights $w^* \in \RR^d$, returns the minimizer $x^* = \arg\min_{x \in \cP} \ip{w^*, x}$ using at most $O(k\log k\log |\cP|)$ comparisons.

*Free variables in the statement:* \cP, d, k, w^*, x^*

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**Your task:** Prove the statement above, using only the definitions and assumptions provided. State any additional standard background results you invoke.