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In this work, we introduce the concept of an entirely new circuit architecture based on the novel, physics-inspired computing paradigm: Memcomputing. In particular, we focus on digital memcomputing machines (DMMs) that can be designed leveraging properties of non-linear dynamical systems; ultimate descriptors of electronic circuits. The working principle of these systems relies on the ability of currents and voltages of the circuit to self-organize in order to satisfy mathematical relations. In particular for this work, we discuss self-organizing gates, namely Self-Organizing Algebraic Gates (SOAGs), aimed to solve linear inequalities and therefore used to solve optimization problems in Integer Linear Programming (ILP) format. Unlike conventional IØgates, SOAGs are terminal-agnostic, meaning each terminal handles a superposition of input and output signals. When appropriately assembled to represent a given ILP problem, the corresponding self-organizing circuit converges to the equilibria that express the solutions to the problem at hand. Because DMM's components are non-quantum, the ordinary differential equations describing it can be efficiently simulated on our modern computers in software, as well as be built in hardware with off-of-the-shelf technology. As an example, we show the performance of this novel approach implemented as Software as a Service (MemCPU XPC) to address an ILP problem. Compared to today's best solution found using a world renowned commercial solver, MemCPU XPC brings the time to solution down from 23 hours to less than 2 minutes.