A Method for Transforming Non-Convex Optimization Problem to Distributed Form
Abstract
:1. Introduction
1.1. Related Work
1.2. Contribution
- The proposed method can be applied to any optimization problem with non-convex separable objective function and coupling constraints.
- Its convergence rate is equivalent to the convergence of Newton’s method with respect to oracle calls.
- Decision variables, cost and constraint functions are not exchanged between agents.
1.3. Paper Organization
1.4. Notations
2. Problem Statement
- The auxiliary problem has the same solution as (1);
- Methods of gradient descent, gradient projection and quasi-Newton methods will operate as distributed optimization methods when applied to the auxiliary problem without any modification (1):
3. Optimization Problem with Equality Constraints
- 1.
- Problem (2) is feasible if and only if problem (7) is feasible.
- 2.
- A pair is the solution to problem (7) if and only if is the solution to (2).
4. Gradient Descent in Variables
Algorithm 1 Gradient descent method (for agent ℓ) |
Input: , , . Output: Algorithm steps: Step 1. Set and ; Step 2. Obtain from neighboring agents; Step 3. Solve problem (11) and (12) for . Let be the corresponding primal and dual solutions; Step 4. Obtain from neighboring agents; Step 5. If Step 6. Calculate : Step 7. Set and go to Step 2; Step 8. Stop: is an -stationary point of problem (2). |
5. Problem with Equality and Inequality Constraints
- 1.
- Problem (19) is feasible if problem (1) is feasible.
- 2.
- Triplet is the solution to problem (19) if and only if is the solution to (1).
6. Newton’s Method
- 1.
- ;
- 2.
- for ;
- 3.
- for ;
- 4.
- For all is solution of the following optimization problem
- functions f, g and h are twice differentiable in some neighborhood of the solution of problem (1), and their second derivatives are Lipshitz-continuous in the neighborhood of the .
- the constraints’ gradients are linear independent in the optimum (linear independence constraint qualification);
- solution has unique corresponding dual variables and in problem (1);
- for , and we have
Algorithm 2 Newton’s method |
Input: f,g,L, . Output: Algorithm steps: Step 1. Set ; Step 2. For each set as a solution of (37) using gradient descent; Step 3. Set and ; Step 4. Set ; Step 5. Solve optimization problem (39) using the gradient descent method; Step 6. Assign Step 7. For all inactive constraints , solve problem (37) using gradient descent and assign its solution to ; Step 8. If , then set and go back to step 5; Step 9. Stop: is an -stationary point. |
7. Application of the Arrow–Hurwicz Algorithm to Problems with Inequality Constraints
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Khamisov, O.O.; Khamisov, O.V.; Ganchev, T.D.; Semenkin, E.S. A Method for Transforming Non-Convex Optimization Problem to Distributed Form. Mathematics 2024, 12, 2796. https://doi.org/10.3390/math12172796
Khamisov OO, Khamisov OV, Ganchev TD, Semenkin ES. A Method for Transforming Non-Convex Optimization Problem to Distributed Form. Mathematics. 2024; 12(17):2796. https://doi.org/10.3390/math12172796
Chicago/Turabian StyleKhamisov, Oleg O., Oleg V. Khamisov, Todor D. Ganchev, and Eugene S. Semenkin. 2024. "A Method for Transforming Non-Convex Optimization Problem to Distributed Form" Mathematics 12, no. 17: 2796. https://doi.org/10.3390/math12172796