Abstract In this paper we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints, are locally smooth. For solving this problem, we propose a linearized quadratic penalty method, i.e., we linearize the objective function and the functional constraints in the penalty formulation at the current iterate and add a quadratic regularization, thus yielding a subproblem that is easy to solve, and whose solution is the next iterate. Under a new adaptive regularization parameter choice, we provide convergence guarantees for the iterates of this method to an $$\epsilon $$ ϵ first-order optimal solution in $${\mathcal {O}}({\epsilon ^{-2.5}})$$ O ( ϵ - 2.5 ) iterations. Finally, we show that when the problem data satisfy Kurdyka–Lojasiewicz property, e.g., are semialgebraic, the whole sequence generated by the proposed algorithm converges and we derive improved local convergence rates depending on the KL parameter. We validate the theory and the performance of the proposed algorithm by numerically comparing it with some existing methods from the literature.

Wastewater treatment processes are used to reduce the amount of polluting substances in waste water resulting from human or industrial consumption. Afterwards, the water is discharged in lakes, rivers, seas, therefore it is important that these systems have an increased efficiency when it comes to treating the waste water. In this paper, we consider a nonlinear wastewater treatment system with four states and one input. We introduce an optimal control problem with state - input constraints, and a corresponding model predictive control scheme and we solve it using ACADO Toolkit. Our main reasons for using the ACADO Toolkit are that it is Open Source, it has a user friendly MATLAB interface and has the advantage of being self - contained (only needs a C++ compiler). We also consider a model predictive control scheme based on the linearization of the nonlinear system at each step, and we solve it using a standard quadratic program solver. From simulations we observe that both approaches produce a similar closed loop behavior, but the linearization based approach is faster in terms of CPU time.

In this paper we consider optimization problems with stochastic composite objective function subject to (possibly) infinite intersection of constraints. The objective function is expressed in terms of expectation operator over a sum of two terms satisfying a stochastic bounded gradient condition, with or without strong convexity type properties. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets, in this paper we consider that each constraint set is given as the level set of a convex but not necessarily differentiable function. Based on the flexibility offered by our general optimization model we consider a stochastic subgradient method with random feasibility updates. At each iteration, our algorithm takes a stochastic proximal (sub)gradient step aimed at minimizing the objective function and then a subsequent subgradient step minimizing the feasibility violation of the observed random constraint. We analyze the convergence behavior of the proposed algorithm for diminishing stepsizes and for the case when the objective function is convex or has a quadratic functional growth, unifying the nonsmooth and smooth cases. We prove sublinear convergence rates for this stochastic subgradient algorithm, which are known to be optimal for subgradient methods on this class of problems. When the objective function has a linear least-square form and the constraints are polyhedral, it is shown that the algorithm converges linearly. Numerical evidence supports the effectiveness of our method in real problems.

We propose a stochastic gradient framework for solving stochastic composite convex optimization problems with (possibly) infinite number of linear inclusion constraints that need to be satisfied almost surely. We use smoothing and homotopy techniques to handle constraints without the need for matrix-valued projections. We show for our stochastic gradient algorithm $\mathcal{O}(\log(k)/\sqrt{k})$ convergence rate for general convex objectives and $\mathcal{O}(\log(k)/k)$ convergence rate for restricted strongly convex objectives. These rates are known to be optimal up to logarithmic factor, even without constraints. We conduct numerical experiments on basis pursuit, hard margin support vector machines and portfolio optimization problems and show that our algorithm achieves state-of-the-art practical performance.