Topological fixed point theory and applications to variational inequalities
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Bibliographic record
Abstract
Abstract This is the first part of a work on generalized variational inequalities and their applications in optimization. It proposes a general theoretical framework for the solvability of variational inequalities with possibly non-convex constraints and objectives. The framework consists of a generic constrained nonlinear inequality ( $\exists\hat{u}\in\Psi(\hat {u})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∃</mml:mi> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>∈</mml:mo> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:math> , $\exists \hat{y}\in\Phi(\hat{u})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∃</mml:mi> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>∈</mml:mo> <mml:mi>Φ</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:math> , with $\varphi(\hat{u},\hat{y},\hat{u})\leq \varphi(\hat{u},\hat{y},v)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>)</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> </mml:math> , $\forall v\in\Psi(\hat{u})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∀</mml:mi> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>ˆ</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:math> ) derived from new topological fixed point theorems for set-valued maps in the absence of convexity. Simple homotopical and approximation methods are used to extend the Kakutani fixed point theorem to upper semicontinuous compact approachable set-valued maps defined on a large class of non-convex spaces having non-trivial Euler-Poincaré characteristic and modeled on locally finite polyhedra. The constrained nonlinear inequality provides an umbrella unifying and extending a number of known results and approaches in the theory of generalized variational inequalities. Various applications to optimization problems will be presented in the second part to this work to be published ulteriorly.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it