We propose a reduced-space formulation for optimizing over trained neural networks where the network's outputs and derivatives are evaluated on a GPU. To do this, we treat the neural network as a "gray box" where intermediate variables and constraints are not exposed to the optimization solver. Compared to the full-space formulation, in which intermediate variables and constraints are exposed to the optimization solver, the reduced-space formulation leads to faster solves and fewer iterations in an interior point method. We demonstrate the benefits of this method on two optimization problems: Adversarial generation for a classifier trained on MNIST images and security-constrained optimal power flow with transient feasibility enforced using a neural network surrogate.

In this paper, we study a latticized optimization problem with fuzzy relational inequality constraints where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno - Weber family of t - norms is considered as fuzzy composition. Sugeno - Weber family of t - norms and t - conorms is one of the most applied one in various fuzzy modelling problems. Thi s family of t - norms and t - conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t - conorms were suggested as addition rules by Sugeno for so - called - fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max - Sugeno - Weber composition and a necessary and sufficient condition is presented for determining the feasibility. Then, based on some theoretical properties of the problem, an algorithm is presented for sol ving this nonlinear problem. It is proved that the algorithm can find the exact optimal solution and an example is presented to illustrate the proposed algorithm.

We compare full-space, reduced-space, and gray-box formulations for representing trained neural networks in nonlinear constrained optimization problems. We test these formulations on a transient stability-constrained, security-constrained alternating current optimal power flow (SCOPF) problem where the transient stability criteria are represented by a trained neural network surrogate. Optimization problems are implemented in JuMP and trained neural networks are embedded using a new Julia package: MathOptAI.jl. To study the bottlenecks of the three formulations, we use neural networks with up to 590 million trained parameters. The full-space formulation is bottlenecked by the linear solver used by the optimization algorithm, while the reduced-space formulation is bottlenecked by the algebraic modeling environment and derivative computations. The gray-box formulation is the most scalable and is capable of solving with the largest neural networks tested. It is bottlenecked by evaluation of the neural network's outputs and their derivatives, which may be accelerated with a graphics processing unit (GPU). Leveraging the gray-box formulation and GPU acceleration, we solve our test problem with our largest neural network surrogate in 2.5$\times$ the time required for a simpler SCOPF problem without the stability constraint.

A nonlinear neural framework, called the Generalized Hopfield network, is proposed, which is able to solve in a parallel distributed manner systems of nonlinear equations. The method is applied to the general nonlinear optimization problem. We demonstrate GHNs implementing the three most important optimization algorithms, namely the Augmented Lagrangian, Generalized Reduced Gradient and Successive Quadratic Programming methods. The study results in a dynamic view of the optimization problem and offers a straightforward model for the parallelization of the optimization computations, thus significantly extending the practical limits of problems that can be formulated as an optimization problem and which can gain from the introduction of nonlinearities in their structure (eg. The Hopfield computational model is almost exclusively applied to the solution of combinatorially complex linear decision problems (eg.

Many real-world systems often involve physical components or operating environments with highly nonlinear and uncertain dynamics. A number of different control algorithms can be used to design optimal controllers for such systems, assuming a reasonably high-fidelity model of the actual system. However, the assumptions made on the stochastic dynamics of the model when designing the optimal controller may no longer be valid when the system is deployed in the real-world. The problem addressed by this paper is the following: Suppose we obtain an optimal trajectory by solving a control problem in the training environment, how do we ensure that the real-world system trajectory tracks this optimal trajectory with minimal amount of error in a deployment environment. In other words, we want to learn how we can adapt an optimal trained policy to distribution shifts in the environment. Distribution shifts are problematic in safety-critical systems, where a trained policy may lead to unsafe outcomes during deployment. We show that this problem can be cast as a nonlinear optimization problem that could be solved using heuristic method such as particle swarm optimization (PSO). However, if we instead consider a convex relaxation of this problem, we can learn policies that track the optimal trajectory with much better error performance, and faster computation times. We demonstrate the efficacy of our approach on tracking an optimal path using a Dubin's car model, and collision avoidance using both a linear and nonlinear model for adaptive cruise control.

Abstract: There exist a number of negative results ([J), [BR), [KV]) about learning on neural nets in Valiant's model [V) for probably approximately correctlearning ("PAClearning"). These negative results are based on an asymptotic analysis where one lets the number of nodes in the neural net go to infinit.y. Hence this analysis is less adequate forthe investigation of learning on a small fixed neural net.

Purdue University Purdue University Purdue University W. Lafayette, IN. 47907 W. Lafayette, IN. 47907 W. Lafayette, IN. 47907 ABSTRACT A nonlinear neural framework, called the Generalized Hopfield network, is proposed, which is able to solve in a parallel distributed manner systems of nonlinear equations. The method is applied to the general nonlinear optimization problem. We demonstrate GHNs implementing the three most important optimization algorithms, namely the Augmented Lagrangian, Generalized Reduced Gradient and Successive Quadratic Programming methods. The study results in a dynamic view of the optimization problem and offers a straightforward model for the parallelization of the optimization computations, thus significantly extending the practical limits of problems that can be formulated as an optimization problem and which can gain from the introduction of nonlinearities in their structure (eg. The ability of networks of highly interconnected simple nonlinear analog processors (neurons) to solve complicated optimization problems was demonstrated in a series of papers by Hopfield and Tank (Hopfield, 1984), (Tank, 1986).

Purdue University Purdue University Purdue University W. Lafayette, IN. 47907 W. Lafayette, IN. 47907 W. Lafayette, IN. 47907 ABSTRACT A nonlinear neural framework, called the Generalized Hopfield network, is proposed, which is able to solve in a parallel distributed manner systems of nonlinear equations. The method is applied to the general nonlinear optimization problem. We demonstrate GHNs implementing the three most important optimization algorithms, namely the Augmented Lagrangian, Generalized Reduced Gradient and Successive Quadratic Programming methods. The study results in a dynamic view of the optimization problem and offers a straightforward model for the parallelization of the optimization computations, thus significantly extending the practical limits of problems that can be formulated as an optimization problem and which can gain from the introduction of nonlinearities in their structure (eg. The ability of networks of highly interconnected simple nonlinear analog processors (neurons) to solve complicated optimization problems was demonstrated in a series of papers by Hopfield and Tank (Hopfield, 1984), (Tank, 1986).

Purdue University W. Lafayette, IN. 47907 ABSTRACT A nonlinear neural framework, called the Generalized Hopfield network, is proposed, which is able to solve in a parallel distributed manner systems of nonlinear equations. The method is applied to the general nonlinear optimization problem. We demonstrate GHNs implementing the three most important optimization algorithms, namely the Augmented Lagrangian, Generalized Reduced Gradient and Successive Quadratic Programming methods. The study results in a dynamic view of the optimization problem and offers a straightforward model for the parallelization of the optimization computations, thus significantly extending the practical limits of problems that can be formulated as an optimization problem and which can gain from the introduction of nonlinearities in their structure (eg. The ability of networks of highly interconnected simple nonlinear analog processors (neurons) to solve complicated optimization problems was demonstrated in a series of papers by Hopfield and Tank (Hopfield, 1984), (Tank, 1986).