Research Challenge: Linear optimization via comparisons and conic dimension
Source paper: Combinatorial Optimization using Comparison Oracles
Reference: arXiv: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^*
Your task: Prove the statement above, using only the definitions and assumptions provided. State any additional standard background results you invoke.