Lexicographic Multi-Objective GP

• Lexicographic multi-objective GP is a framework that optimizes multiple posynomial objectives in a strict priority order.
• It employs logarithmic transformation to reveal convexity, allowing the use of strong duality and efficient optimization techniques.
• The approach uses a recursive algorithm that solves prioritized subproblems, ensuring feasibility and unique solutions at each level.

Lexicographic Multi-Objective Geometric Programming (LGP) extends classical geometric programming (GP) to problems with multiple posynomial objectives arranged in a strict priority order. Each objective function and constraint is a posynomial—i.e., a sum of monomials with positive coefficients and real exponents—over positive decision variables. The lexicographic framework seeks a solution that is optimal in the first objective, then among all such solutions is optimal in the second objective, and so forth, capturing a precise notion of priority among objectives. The convexity of logarithmic-transformed posynomial GP problems enables the application of strong duality and efficient optimization techniques (0912.1832).

1. Mathematical Formulation of Lexicographic Multi-Objective GP

Let $x = (x_1, \ldots, x_n) \in \mathbb{R}_{++}^n$ denote the vector of strictly positive decision variables, and consider $p$ posynomial objectives: $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$ subject to $m$ posynomial inequality constraints: $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$ The lexicographic multi-objective GP is: $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$ Here, $\lexmin$ denotes that $f_1(x)$ is minimized first; within its optimizer set, $f_2(x)$ is minimized, and so on (0912.1832).

2. Well-Definedness, Feasibility, and Existence Conditions

Three essential conditions guarantee the existence and uniqueness of solutions at each priority level:

• Non-emptiness: The feasible set $\{x > 0 : g_i(x) \leq 1\,\, \forall i\}$ must be nonempty.
• Reduction at Levels: At rank $p$0, introducing additional constraints $p$1, the modified feasible set must remain nonempty.
• Unique Solvability: The exponent matrix of the active monomials at each subproblem must have full column rank $p$2. This condition ensures strong duality and unique solutions as discussed in Duffin–Peterson–Zener theory.

If any feasible set becomes empty under the addition of higher-priority constraints, the lexicographic chain of subproblems fails to yield a solution (0912.1832).

3. Duality Theory for Lexicographic GP

For a single-posynomial GP, the dual is constructed with variables $p$3 associated to each monomial term:

$p$4

where $p$5 and $p$6 are the monomial coefficients and exponents, respectively. The unique solution $p$7 can be recovered using

$p$8

together with term-matching conditions relating monomial values and optimal dual variables.

In the lexicographic setting, the dualization proceeds recursively. At level $p$9, new dual variables are introduced for each monomial in the current objective and all additional constraints inherited from higher priorities. The enlarged exponent matrix must satisfy the normality and orthogonality conditions across all levels. Weak duality guarantees $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$0 for each level; strong duality (zero duality gap) holds under strict feasibility and full-rank conditions (0912.1832).

4. Algorithmic Procedure

The standard algorithm for lexicographic GP, given priority ordering $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$1:

1. Step 1: Solve the single-objective GP subproblem $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$2 s.t. $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$3. Outcome: $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$4, $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$5.
2. Step 2: Solve $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$6 s.t. $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$7. Outcome: $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$8, $$f_k(x) = \sum_{t=1}^{T_k} c_{kt} \prod_{j=1}^n x_j^{a_{ktj}}, \quad c_{kt} > 0, \; a_{ktj} \in \mathbb{R},\;\; k=1, \ldots, p,$$9.
3. Step $m$0: Solve $m$1 s.t. $m$2.
4. Repeat until $m$3; the solution $m$4 is the lexicographic optimum.

Each level may be solved via the dual problem, leveraging convex GP solvers (interior-point or steepest-descent methods). The approach guarantees finite convergence, as each subproblem is individually convex. The computational complexity increases with the number of objectives $m$5 and total monomial terms (0912.1832).

5. Illustrative Numerical Example

From (0912.1832), consider $m$6, $m$7 with objectives and constraints:

• Primal Level 1: $m$8, subject to $m$9, $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$0.
• Dual formation: Variables $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$1 (for $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$2), $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$3 (for two constraints). Constraints on dual variables:

$$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$4

• Solution:

$$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$5, $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$6, $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$7, $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$8, $$g_i(x) = \sum_{t=1}^{T_i} d_{it} \prod_{j=1}^{n} x_j^{b_{itj}} \le 1,\,\, d_{it} > 0,\, b_{itj} \in \mathbb{R},\;\; i=1,\ldots, m.$$9; Dual optimum $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$0, Primal optimum: $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$1, $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$2.

• Primal Level 2: $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$3, s.t. $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$4, $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$5.
• Solving the extended dual yields $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$6. The tuple $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$7 and corresponding $\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$8 comprise the lexicographic optimum.

6. Extensions, Limitations, and Open Problems

• Sensitivity to Priority Assignment: The ordering of objective priorities critically affects the solution. Techniques such as the Analytic Hierarchy Process (AHP) may be employed for priority selection before applying LGP.
• Multiple Equal Priorities: If two or more objectives have equal rank, a concurrent multi-objective GP (using Pareto optimality) is appropriate.
• High Degree of Difficulty: As the total number of monomials increases far beyond the number of variables ($\begin{aligned} \lexmin\quad &(f_1(x), f_2(x), \ldots, f_p(x)) \ \text{s.t.}\quad &g_i(x) \leq 1,\,\; i=1,\ldots,m, \ &x \in \mathbb{R}_{++}^n. \end{aligned}$9), the dual system grows in complexity. Computationally efficient interior-point methods remain necessary.
• Extensions: Fuzzy lexicographic GP, accommodating fuzzy objective priorities, and dynamic lexicographic GP for time-variant systems represent proposed extensions.
• Open Problems: Incorporation of coefficient uncertainty (for $\lexmin$0), hybridizing lexicographic and ε-constraint or weighted-sum approaches, and the development of parallel algorithms for large-scale instances remain open research areas (0912.1832).