Nonconvex Variables Research Articles

Smart Energy Borrowing and Relaying in Wireless-Powered Networks: A Deep Reinforcement Learning Approach

Wireless energy harvesting (EH) communication has long been considered a sustainable networking solution. However, it has been limited in efficiency, which has been a major obstacle. Recently, strategies such as energy relaying and borrowing have been explored to overcome these difficulties and provide long-range wireless sensor connectivity. In this article, we examine the reliability of a wireless-powered communication network by maximizing the net bit rate. To accomplish our goal, we focus on enhancing the performance of hybrid access points and information sources by optimizing their transmit power. Additionally, we aim to maximize the use of harvested energy, by using energy-harvesting relays for both information transmission and energy relaying. However, this optimization problem is complex, as it involves non-convex variables and requires combinatorial relay selection indicator optimization for decode and forward (DF) relaying. To simplify this problem, we utilize the Markov decision process and deep reinforcement learning framework based on the deep deterministic policy gradient algorithm. This approach enables us to tackle this non-tractable problem, which conventional convex optimization techniques would have difficulty solving in complex problem environments. The proposed algorithm significantly improved the end-to-end net bit rate of the smart energy borrowing and relaying EH system by 13.22%, 27.57%, and 14.12% compared to the benchmark algorithm based on borrowing energy with an adaptive reward for Quadrature Phase Shift Keying, 8-PSK, and 16-Quadrature amplitude modulation schemes, respectively.

Wireless UAV Rotary Wing Communication with Ground Nodes Using Successive Convex Approximation and Energy Saving Mode

Communication with several ground nodes (GNs) is enabled through wireless communications by means of a rotary-wing unmanned aerial vehicle (UAV). For every GN, the requirements of communication throughput is satisfied while minimizing the communication and propulsion related energy thereby reducing the total energy consumption of the UAV. The total completion time of the mission, the GN time allocation for communication as well as the trajectory of UAV are optimized jointly to formulate the problem of energy minimization. The UAV rotary wing model of power consumption for propulsion power in a closed-form is also derived. The complexity of the problem increases with the involvement of several variables infinitely over time and the non-convex variables, making it challenging to be solved optimally. The hovering location sets are visited by the UAV and at each location, communication is established with the corresponding GNs. This simplified design for fly-hover-communicate is established for tackling the underlying issue. Convex optimization techniques and travelling salesman problem with neighborhood (TSPN) are leveraged for optimization of the trajectory that connects the hovering locations and the duration of hovering by means of the proposed algorithm design. Communication of the UAV during operation is considered as a general scenario. Discretization of the problem to its equivalent is done in the proposed novel path discretization technique by means of optimized variables of finite number. Further, a successive convex approximation (SCA) scheme is applied for obtaining a solution that is locally optimal. A comparison of the proposed design is performed with the benchmark schemes and it is found that they are outperformed based on numerical results.

Efficient Global Optimal Resource Allocation in Non-Orthogonal Interference Networks

Many resource allocation tasks are challenging global (i.e., non-convex) optimization problems. The main issue is that the computational complexity of these problems grows exponentially in the number of variables instead of polynomially as for many convex optimization problems. However, often the non-convexity stems only from a subset of variables. Conventional global optimization frameworks like monotonic optimization or DC programming treat all variables as global variables and require complicated, problem specific decomposition approaches to exploit the convexity in some variables. To overcome this challenge, we develop an easy-to-use algorithm that inherently differentiates between convex and non-convex variables, preserving the low computational complexity in the number of convex variables. Another issue with these widely used frameworks is that they may suffer from severe numerical problems. We discuss this issue in detail and provide a clear motivating example. The solution to this problem is to replace the traditional approach of finding an {\epsilon}-approximate solution by the novel concept of {\epsilon}-essential feasibility. The underlying algorithmic approach is called successive incumbent transcending (SIT) algorithm and builds the foundation of our developed algorithm. A further highlight of this algorithm is that it inherently treats fractional objectives making the use of Dinkelbach's iterative algorithm obsolete. Numerical experiments show a speed-up of four orders of magnitude over state-of-the-art algorithms and almost three orders of magnitude of additional speed-up over Dinkelbach's algorithm for fractional programs.

Assessment of a non-adaptive deterministic global optimization algorithm for problems with low-dimensional non-convex subspaces

The optimum and at least one optimizing point for convex nonlinear programs can be approximated well by the solution to a linear program (a fact long used in branch and bound algorithms). In more general problems, we can identify subspaces of ‘non-convex variables’ such that, if these variables have sufficiently small ranges, the optimum and at least one optimizing point can be approximated well by the solution of a single linear program. If these subspaces are low-dimensional, this suggests subdividing the variables in the subspace a priori, then producing and solving a fixed, known number of linear programs to obtain an approximation to the solution. The total amount of computation is much more predictable than that required to complete a branch and bound algorithm, and the scheme is ‘embarrassingly parallel’, with little need for either communication or load balancing. We compare such a non-adaptive scheme experimentally to our GlobSol branch and bound implementation, on those problems from the COCONUT project Lib1 test set with non-convex subspaces of dimension four or less, and we discuss potential alterations to both the non-adaptive scheme and our branch and bound process that might change the scope of applicability.

A general framework for convexity analysis in deterministic global optimization

In previous work, we, and also Epperly and Pistikopoulos, proposed an analysis of general nonlinear programs that identified certain variables as convex, not ever needing subdivision, and non-convex, or possibly needing subdivision in branch and bound algorithms. We proposed a specific algorithm, based on a generated computational graph of the problem, for identifying such variables. In our previous work, we identified only independent variables in the computational graph. Here, we examine alternative sets of non-convex variables consisting not just of independent variables, but of a possibly smaller number of intermediate variables. We do so with examples and theorems. We also apply variants of our proposed analysis to the well-known COCONUT Lib-1 test set. If the number of such non-convex variables is sufficiently small, it may be possible to fully subdivide them before analysis of ranges of objective and constraints, thus dispensing totally with the branch and bound process. Advantages to such a non-adaptive process include higher predictability and easier parallizability. We present an algorithm and exploratory results here, with a more complete empirical study in a subsequent paper.

Continuous global optimization of structured process systems models

Nonconvex models for the optimization of process systems in chemical engineering give rise to multiple suboptimal solutions, and a number of complications that often cause failure of standard local optimization techniques. A deterministic branch and bound algorithm is presented in this paper for the global optimization of structured process systems models that include non-convexities introduced by concave univariate, bilinear and linear fractional terms. The proposed branch and contract algorithm relies on a bounds contraction operation, which consists of the solution of a finite sequence of convex bounds-contraction subproblems for the subset of nonconvex variables in a problem. The application of the proposed algorithm is illustrated with several numerical examples, which include heat exchanger networks, chemical reactors, simplified process flowsheets, and waste-water treatment systems. The results show that by executing the contraction operation at selected branch and bound nodes, large portions of the search region over which the objective function takes only values above a known upper bound are eliminated. It is shown that with the proposed approach the total number of nodes in the solution tree is kept relatively small, and for some problems, no branching is required at all.

Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.

On nonconvex optimization problems with separated nonconvex variables

A mathematical programming problem is said to have separated nonconvex variables when the variables can be divided into two groups: x=(x1,...,xn) and y=( y1,...,yn), such that the objective function and any constraint function is a sum of a convex function of (x, y) jointly and a nonconvex function of x alone. A method is proposed for solving a class of such problems which includes Lipschitz optimization, reverse convex programming problems and also more general nonconvex optimization problems.

Convex minimization under Lipschitz constraints

We consider the problem of minimizing a convex functionf(x) under Lipschitz constraintsf i (x)≤0,i=1,...,m. By transforming a system of Lipschitz constraintsf i (x)≤0,i=l,...,m, into a single constraints of the formh(x)-∥x∥2≤0, withh(·) being a closed convex function, we convert the problem into a convex program with an additional reverse convex constraint. Under a regularity assumption, we apply Tuy's method for convex programs with an additional reverse convex constraint to solve the converted problem. By this way, we construct an algorithm which reduces the problem to a sequence of subproblems of minimizing a concave, quadratic, separable function over a polytope. Finally, we show how the algorithm can be used for the decomposition of Lipschitz optimization problems involving relatively few nonconvex variables.