A cutting-plane method learns data manifolds efficiently.
problem Classifying data manifolds with continuous parameters.
method Iterative algorithm M_{CP} based on cutting-plane approach solving a quadratic semi-infinite programming problem.
result M_{CP} provides superior generalization performance compared to conventional methods.
We present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems, and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of a…
Develops new reinforcement learning methods for complex constrained decision-making problems.
problem Complex constrained decision-making problems with a continuum of constraints.
method Proposes semi-infinitely constrained Markov decision processes (SICMDPs) and two reinforcement learning algorithms: SI-CRL and SI-CPO.
result Demonstrates the effectiveness of SI-CRL and SI-CPO in solving complex sequential decision-making tasks.
Develops exact convex optimization formulations for neural networks.
problem Training two-layer neural networks with rectified linear units.
method Uses semi-infinite duality and minimum norm regularization to develop exact convex optimization formulations.
result Shows equivalence of ReLU networks trained with weight decay to block ℓ1 penalized convex models. Neural networks can find financial arbitrage opportunities without needing market models.
problem Finding arbitrage opportunities in financial markets without using market models.
method Used neural networks to solve convex semi-infinite programs and detect arbitrage opportunities.
result Neural networks can detect model-free static arbitrage strategies in financial markets.
We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi-infinite programming problem, tha…
Neural networks solve copositive programs, revealing insights into training problems.
problem Training two-layer vector-output ReLU neural networks.
method Convex analysis and copositive programming.
result Neural networks solve copositive programs, providing insights into training problems.
Polynomial-time convex optimization for CNNs with ReLU activations.
problem Training Convolutional Neural Networks (CNNs) with ReLU activations.
method Developed a convex analytic framework using semi-infinite duality to formulate equivalent convex optimization problems for CNN architectures.
result Proved that two-layer CNNs can be globally optimized via an ℓ2 norm regularized convex program. New method solves large-scale QCPs using low-discrepancy sequences.
problem Solving large-scale Quadratically Constrained Quadratic Programs (QCQP).
method Transforming QCQP into a linear problem via low-discrepancy sampling.
result Approximate solutions converge to true solutions and have finite sample error bounds.
We simplify no-arbitrage bounds calculation for financial derivatives.
problem Calculating robust replication of forward-start straddles from market data.
method Proposed a discretisation scheme and a new linear programming approach to the dual problem.
result Reconciled two approaches: semi-infinite linear programming and optimal martingale measures.
Notes on Morse Homology, focusing on gradient flow lines and semi-infinite dimensional cases.
problem Exploring Morse Homology and its applications in semi-infinite dimensional spaces.
method Presentation of concepts in finite dimensional Morse Homology, with an eye towards generalization to semi-infinite dimensions.
result Intuition for Floer homology through finite dimensional Morse Homology concepts.
Novel approximation hierarchy for sparse quadratic programs.
problem Sparse Quadratic Programs with Cardinality Constraints.
method Exploits rank-dominating eigenvectors for min-max optimization over binary variables.
result Efficient screening of nonzero elements with scalable optimization algorithms.
We consider a proximal operator given by a quadratic function subject to bound constraints and give an optimization algorithm using the alternating direction method of multipliers (ADMM). The algorithm is particularly efficient to solve a collection of proximal operators that share the same quadratic form, or if the qu…
Abstract perspective on quadratic programming for optimal portfolio allocation.
problem Optimal allocation problems in long portfolio theory.
method Using maximum principles and distinguished boundaries in reproducing kernel Hilbert spaces.
result Support of an optimal distribution lies in a variety intersecting a distinguished boundary.
Paper presents an ADMM-based approach to efficiently integrate quadratic programming layers into neural networks.
problem Integrating quadratic programs into neural networks for optimization.
method An ADMM-based network layer architecture for solving quadratic programs efficiently.
result The ADMM layer is approximately an order of magnitude faster than existing methods for medium scaled problems.
Paper proposes a robust method for inferring parameters in multiobjective optimization.
problem Uncertainty in hypothetical decision-making problem, data quality, and parameter space.
method Wasserstein distributionally robust approach for inverse multiobjective optimization.
result WRO-IMOP minimizes worst-case expected loss over a Wasserstein ball of distributions.
Eigen-decomposition simplifies quadratic programming with equality constraints.
problem Optimizing solutions under linear equality constraints in quadratic programming.
method Eigenvalue decomposition of the quadratic term matrix to project optimal solutions.
result Established a linear mapping between EQP formulations with and without diagonalized Q. Paper optimizes financial trading strategies under uncertain market conditions.
problem Guaranteeing robust positive expected profits in financial systems.
method Transformed semi-infinite constraints into structured policies and proposed a novel graphical approach.
result Demonstrated superior risk-adjusted returns and downside risk compared to conventional strategies.
Faster algorithms for structured SVMs reduce computation time.
problem Efficiently solving quadratic programming problems with specific structures.
method Designing nearly-linear time algorithms for quadratic programs with low-rank factorizations and few linear constraints.
result First nearly-linear time algorithms for solving quadratic programs with specific structures.
V-matrix method fails to consistently estimate conditional probabilities.
problem Inconsistent solutions in V-matrix method for conditional probability estimation.
method Construct constrained quadratic programming problems with inconsistent inequality constraints.
result V-matrix method may not always have a consistent solution for conditional probability estimation.
New method solves constrained stochastic optimization problems efficiently.
problem Online statistical inference of constrained stochastic nonlinear optimization problems.
method Stochastic Sequential Quadratic Programming (StoSQP) with iterative sketching solver.
result The rescaled primal-dual sequence converges to a mean-zero Gaussian distribution.
Method solves complex optimization problems with high probability bounds.
problem Nonlinear equality constrained stochastic optimization problems.
method Step-search sequential quadratic programming method.
result High-probability bound on iteration complexity for first-order stationarity.
New homology for infinite multi-colored braids, completing previous work.
problem Categorification of highest-weight projectors for infinite braids.
method Defining limiting Khovanov-Rozansky homology for semi-infinite braids.
result Categorifies highest-weight projectors for a large class of braids.
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…
New method improves support estimation for unknown distributions.
problem Estimating the support size of an unknown distribution.
method Regularized Weighted Chebyshev Approximations, joint optimization of bias and variance, linear programming.
result Significant improvements in worst-case risk for synthetic data and accurate bacterial genus estimation for microbiome data.
LCC algorithm maps instances to a central space for better classification.
problem Improving classification accuracy for various datasets.
method Formulated as a quadratic program, simplified to a linear program, uses kernel functions for non-linear cases.
result LCC outperforms other methods in accuracy on standard datasets.
Novel method solves group synchronization with robust corruption tolerance.
problem Group synchronization with high corruption tolerance.
method Quadratic programming formulation exploiting cycle consistency.
result Global minimum recovers corruption levels under mild conditions.
New method solves nonseparable stochastic control problems.
problem Nonseparable and non-monotonic stochastic control problems.
method Scenario-decomposition solution framework using progressive hedging algorithm.
result Extends reach of stochastic optimal control.
The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones polynomial can be computed by a suitable (almost tight) state sum and the solution of a …
Convex optimization refines neural network training, improving model performance and reducing hyperparameter sensitivity.
problem Training deep neural networks using non-convex optimization methods often leads to suboptimal solutions and requires extensive tuning.
method Formulate neural network training as convex programs with regularization terms, leveraging sparse recovery models and semi-infinite programming theory.
result Convex models can achieve global optima and outperform traditional non-convex methods, with improved robustness to hyperparameters.
New method solves optimization problems with stochastic objectives and constraints.
problem Optimization problems with stochastic objectives and deterministic constraints.
method Trust-region interior-point stochastic sequential quadratic programming (TR-IP-SSQP) method.
result Global almost-sure convergence to first-order stationary points under standard assumptions.
We propose a randomized second-order method for optimization known as the Newton Sketch: it is based on performing an approximate Newton step using a randomly projected or sub-sampled Hessian. For self-concordant functions, we prove that the algorithm has super-linear convergence with exponentially high probability, wi…
New algorithm speeds up path computation for optimal models.
problem Finding the exact path of optimal models from a finite set.
method Dynamic programming approach for linear time computation.
result Dynamic programming achieves linear time for breakpoints computation.
In this paper we propose a tractable quadratic programming formulation for calculating the equilibrium term structure of electricity prices. We rely on a theoretical model described in [21], but extend it so that it reflects actually traded electricity contracts, transaction costs and liquidity considerations. Our nume…
New methods for phase estimation in mixed signals, improving source separation.
problem Estimating phases of mixed complex signals from multichannel observations.
method Three approaches: heuristic, alternate minimization, and convex relaxation.
result Convex relaxation approach yields best results, including exact source separation.
Improved understanding of low-rank solutions in SDPs via smoothed analysis.
problem Finding low-rank solutions to semidefinite programs efficiently.
method Penalty function formulation and smoothed analysis to avoid worst-case matrices.
result All approximate local optima are global optima for rank-constrained SDPs under certain conditions.
Efficiently solves heterogeneous QPs by reducing variables using instance-specific projections.
problem Solving high-dimensional quadratic programming problems efficiently.
method Data-driven framework with a graph neural network generating projections tailored to each QP instance.
result Produces high-quality solutions with reduced computation time, outperforming existing methods.
Proposes a new algorithm for nonconvex sparse learning problems that converges quickly.
problem Nonconvex sparse learning problems in high dimensions.
method Combines proximal Newton algorithm with DC programming for multi-stage convex relaxation.
result Achieves quadratic convergence and finds sparse approximate local optima.
New method trains Boltzmann machines without supervision.
problem Training unsupervised learning models.
method Mixed binary quadratic feasibility problem formulation.
result Theory validated on XOR patterns.
The paper tackles control policy learning for unknown systems using convex optimization.
problem Learning control policies for unknown linear dynamical systems to maximize a quadratic reward function.
method Sequential convex programming to optimize expected reward over posterior system parameter distribution.
result The method achieves reliable local convergence and robust stability, demonstrated with strong performance and robustness in simulations and real-world applications.
Neural network discovers exact solutions to QP with linear constraints.
problem Discovering exact solutions to Quadratic Programs (QP) with linear constraints using neural networks.
method Proposes a neural network modeling approach that analytically derives model parameters from problem coefficients, ensuring closed-form solutions without training.
result The closed-form NN model produces exact solutions for every critical region of the QP solution function, outperforming DNNs and commercial solvers in terms of optimality and feasibility.
Paper approximates Kelly betting for wealth growth.
problem Optimizing wealth growth in Kelly betting.
method Taylor-based approximation for quadratic programming.
result Closed-form approximate solution with interesting properties.
Paper addresses adversarial robustness in deep learning.
problem Fragility of deep learning to adversarial perturbations.
method Semi-infinite constrained learning and non-convex duality theory.
result Adversarial training is equivalent to a statistical problem over perturbation distributions.
New algorithm tackles stochastic optimization with inequality constraints.
problem Stochastic optimization with inequality constraints in various applications.
method Active-set stochastic sequential quadratic programming (StoSQP) with a differentiable exact augmented Lagrangian.
result Global convergence for any initialization, KKT residuals converge to zero almost surely.
Lasso method applied to polynomial models with hierarchy constraints.
problem Estimating parameters in polynomial models with hierarchy constraints.
method Using lasso and standard quadratic programming techniques to estimate parameters.
result The proposed methodology outperforms existing techniques in terms of validation error and model size.
The paper shows how label noise in training can lead to solutions that solve a Lasso program.
problem Understanding the implicit bias of training algorithms in overparametrised models.
method Analyzing the continuous time version of the training dynamics of a quadratically parametrised model.
result The stochastic flow implicitly solves a Lasso program, providing convergence guarantees and support recovery conditions.
The paper tackles denoising of function samples modulo 1.
problem Recover smooth estimates of a function's modulo 1 samples from noisy data.
method Formulates and solves a quadratically constrained quadratic program relaxation.
result Demonstrates robustness of the approach to noise.
A new portfolio optimization model minimizes maximum drawdown, offering faster and more robust solutions.
problem Optimizing portfolios during financial distress, especially during crises.
method Linearization of Markowitz model based on maximum drawdown, with a Mixed-Integer Linear Programming variation.
result 200 times faster solving time with a more profitable and robust solution.