A Unified Approach for Maximizing Continuous $\gamma$-weakly DR-submodular Functions

\(\) This paper presents a unified approach for maximizing continuous \(\gamma\)-weakly DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the convex feasible region. We consider settings where the oracle provides access to either … Read more

Doubly stochastic primal dual splitting algorithm with variance reduction for saddle point problems

The (structured) saddle-point problem involving the infimal convolution in real Hilbert spaces finds applicability in many applied mathematics disciplines. For this purpose, we develop a stochastic primal-dual splitting (PDS) algorithm with loopless variance-reduction (VR) for solving this generic problem. A PDS algorithm aims to overcome the well-known shortcomings of common splitting methods by solving the … Read more

Almost-sure convergence of iterates and multipliers in stochastic sequential quadratic optimization

Stochastic sequential quadratic optimization (SQP) methods for solving continuous optimization problems with nonlinear equality constraints have attracted attention recently, such as for solving large-scale data-fitting problems subject to nonconvex constraints. However, for a recently proposed subclass of such methods that is built on the popular stochastic-gradient methodology from the unconstrained setting, convergence guarantees have been … Read more

Asynchronous Iterations in Optimization: New Sequence Results and Sharper Algorithmic Guarantees

We introduce novel convergence results for asynchronous iterations that appear in the analysis of parallel and distributed optimization algorithms. The results are simple to apply and give explicit estimates for how the degree of asynchrony impacts the convergence rates of the iterates. Our results shorten, streamline and strengthen existing convergence proofs for several asynchronous optimization … Read more

A Stochastic-Gradient-based Interior-Point Algorithm for Solving Smooth Bound-Constrained Optimization Problems

A stochastic-gradient-based interior-point algorithm for minimizing a continuously differentiable objective function (that may be nonconvex) subject to bound constraints is presented, analyzed, and demonstrated through experimental results. The algorithm is unique from other interior-point methods for solving smooth (nonconvex) optimization problems since the search directions are computed using stochastic gradient estimates. It is also unique … Read more

Multi-model Partially Observable Markov Decision Processes

We propose a new multi-model partially observable Markov decision process (MPOMDP) model to address the issue of model ambiguity in partially observable Markov decision process. Here, model ambiguity is defined as the case where there are multiple credible optimization models with the same structure but different model parameters. The proposed MPOMDP model aims to learn … Read more

Sequential Quadratic Optimization for Stochastic Optimization with Deterministic Nonlinear Inequality and Equality Constraints

A sequential quadratic optimization algorithm for minimizing an objective function defined by an expectation subject to nonlinear inequality and equality constraints is proposed, analyzed, and tested. The context of interest is when it is tractable to evaluate constraint function and derivative values in each iteration, but it is intractable to evaluate the objective function or … Read more

Data-Driven Stochastic Dual Dynamic Programming: Performance Guarantees and Regularization Schemes

We propose a data-driven scheme for multistage stochastic linear programming with Markovian random parameters by extending the stochastic dual dynamic programming (SDDP) algorithm. In our data-driven setting, only a finite number of historical trajectories are available. The proposed SDDP scheme evaluates the cost-to-go functions only at the observed sample points, where the conditional expectations are … Read more

A Simplified Convergence Theory for Byzantine Resilient Stochastic Gradient Descent

In distributed learning, a central server trains a model according to updates provided by nodes holding local data samples. In the presence of one or more malicious servers sending incorrect information (a Byzantine adversary), standard algorithms for model training such as stochastic gradient descent (SGD) fail to converge. In this paper, we present a simplified … Read more

Statistical performance of subgradient step-size update rules in Lagrangian relaxations of chance-constrained optimization models

Lagrangian relaxation schemes, coupled with a subgradient procedure, are frequently employed to solve chance-constrained optimization models. The subgradient procedure typically relies on a step-size update rule. Although there is extensive research on the properties of these step-size update rules, there is little consensus on which rules are most suited in practice. This is especially so … Read more