Multi-Objective LQG Design with Primal-Dual Method
Abstract
:1. Introduction
2. Finite-Horizon LQG Problem
- for all and ;
- , .
3. Main Results
3.1. Lagrangian Solution
- Primal feasibility condition:
- Complementary slackness condition:
- Dual feasibility condition:
- Stationary condition :
- If the strict inequality is already satisfied with obtained using the standard Riccati equation, then solves the complementary slackness condition. We do not need to do anything in this case.
- Moreover, if the equality is satisfied with obtained using the standard Riccati equation, then any solves the complementary slackness condition. However, when , the corresponding may be different from . Therefore, to use the variables obtained in Proposition 1 as a solution to the KKT condition, we need to set .
- Lastly, assume that holds with obtained using the standard Riccati equation. Then, some solves the complementary slackness condition if . Suppose that is such a number. Then, the corresponding tuple satisfies the KKT condition.
- 1.
- f is continuous over ;
- 2.
- holds for any ;
- 3.
- If holds for some , then ;
- 4.
- ;
- 5.
- There exists a such that ;
- 6.
- Define the set-valued mapping . Then, is a closed line segment.
- From the definition, is linear in , is rational, whose entries are finite for a finite because the inverse matrix in is finite for all . Therefore, from the definition, is also rational and finite over , which implies that is continuous in . This completes the proof.
- We only need to prove the inequality for . By contradiction, suppose that holds. For a fixed , we see from the KKT condition that the problem is nothing but the optimization
- The fourth statement is true due to Assumption 3.
- Note that the objective in (13) can be replaced with without changing the optimal solutions. As , the objective converges to , which implies as due to the strict feasibility assumption in Assumption 2. Since f is continuous over from the first statement, there should exists such that .
- Define and . From the continuity of f, the supremum and infimum are attained; otherwise, f should be discontinuous. Therefore, we can define and . From the second statement, we see that for all . It completes the proof.
3.2. Algorithm
Algorithm 1 Primal-Dual Method with Bisection Line Search. |
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Algorithm 2 Policy evaluation . |
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3.3. Suboptimality
3.4. Computational Efficiency
3.5. Deterministic Cases
3.6. Example
4. Conclusions
Funding
Conflicts of Interest
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Lee, D. Multi-Objective LQG Design with Primal-Dual Method. Mathematics 2023, 11, 1857. https://doi.org/10.3390/math11081857
Lee D. Multi-Objective LQG Design with Primal-Dual Method. Mathematics. 2023; 11(8):1857. https://doi.org/10.3390/math11081857
Chicago/Turabian StyleLee, Donghwan. 2023. "Multi-Objective LQG Design with Primal-Dual Method" Mathematics 11, no. 8: 1857. https://doi.org/10.3390/math11081857
APA StyleLee, D. (2023). Multi-Objective LQG Design with Primal-Dual Method. Mathematics, 11(8), 1857. https://doi.org/10.3390/math11081857